CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

eSolutions Manual - Powered by Cognero Page 7

2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

eSolutions Manual - Powered by Cognero Page 8

2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

eSolutions Manual - Powered by Cognero Page 13

2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

eSolutions Manual - Powered by Cognero Page 15

2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

eSolutions Manual - Powered by Cognero Page 16

2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

eSolutions Manual - Powered by Cognero Page 18

2-1 Relations and Functions

CCSS STRUCTURE  State the domain and range of each relation. Then determine whethereach relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

1. 

SOLUTION:  The left side of  the mapping is the domain and the right side is the range.The members of the domain are the x-values of the relation while the members ofthe range are the y-values.   D = {–2, 5, 6}, R = { –8, 1, 3};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.

2. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4}, R = {–1, 2, 3, 5};   The relation is not a function because 1 is mapped to both 2 and 5.

3. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–2, 1, 4, 8}, R = {–4, –2, 6};   Each element of the domain is paired with exactly one element of the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

4. BASKETBALL The table shows the average points per game for Dwayne Wade of the Miami Heat for four years.

a. Assume that the ages are the domain. Identify the domain and range. b. Write a relation of ordered pairs for the data. c. State whether the relation is discrete or continuous. d. Graph the relation. Is this relation a function?  

SOLUTION:  a. Since the ages are the domain, the average points per game are the range. D = {24, 25, 26, 27}, R = {24.6, 27.2, 27.4, 30.2}   b. In writing ordered pairs for the relation, the members of the domain are the x-values and the members of the range are the y -values. {(24, 27.2), (25, 27.4), (26, 24.6), (27, 30.2)}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function as each element of the domain is paired with exactly one element of the range.  

Graph each equation, and determine the domainand range. Determine whether the equation is afunction, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.  

5. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The graph of the function is a line. So the function is continuous.

x y = 5x + 4 0 4 1 5 2 14 3 19

-1 -1 -2 -6 -3 -11

6. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range correspond to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed using a straight line. So the relation is continuous.

x y = -4x - 2 0 -2 1 -6 2 -10 3 -14

-1 2 -2 6 -3 10

7. 

SOLUTION:  To graph the equation, substitute different values of xin the equation and solve for y . Then connect the points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the graph is a function.   The function is neither one-to-one nor onto becausethe elements in the domain do not have unique images and the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed using a smooth curve. Sothe relation is continuous.

x y = 3x2

0 0 1 3 2 12 3 27

-1 3 -2 12 -3 27

8. 

SOLUTION:  The graph of the equation is a vertical line through (7,0). 

  In this equation x is always 7 for any value of y . D = {7}; R = {all real numbers};   The only element in the domain is mapped to all the elements in the range. So it is not a function.   The domain has a finite number (1) of elements, so the relation is not continuous.

Evaluate each function. 

9. 

SOLUTION:  

  Replace x by –3.  

10. 

SOLUTION:  

  Replace x with 5.  

State the domain and range of each relation. Then determine whether each relation is a function . If it is a function, determine if it is one-to-one, onto, both, or neither.  

11. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–0.3, 0.4, 1.2}, R = {–6, –3, –1, 4}   1.2 is mapped to both –1 and 4. So the relation is not a function.

12. 

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–8, 2, 4}; R = {–6,  –4, 14};   –8 is mapped to both –4 and 14. So the relation is nota function.

13. {(–3, –4), (–1, 0), (3, 0), (5, 3)}

SOLUTION:  The members of the domain are the x-values of the relation while the members of the range are the y-values.   D = {–3, –1, 3, 5}; R = {–4, 0, 3}   Each element of the domain is paired with exactly one element in the range. So, the relation is a function.   The function is onto because each element of the range corresponds to an element of the domain.

14. POLITICS The table below shows the population ofseveral states and the number of U.S. representatives from those states. a. Make a graph of the data with population on the horizontal axis and representatives on the vertical axis. b. Identify the domain and range. c. Is the relation discrete or continuous? d. Does the graph represent a function? Explain yourreasoning.

SOLUTION:  a. Scale each axis of the graph by 5. Since populationis on the horizontal axis, these are the x-values of therelation or the domain. The number of representatives is the range. Plot the data from the table as ordered pairs on the graph.  

  b. The members of the domain are the x-values of the relation while the members of the range are the y-values.  D = {8.07, 12.44, 16.03, 19.00, 20.90, 33.93}; R={13, 19, 25, 29, 32, 53}   c. The domain is a set of individual points. So the relation is discrete.   d. The relation is a function because each domain value is paired with only one range value.

CCSS STRUCTURE  Graph each equation, and determine the domain and range. Determine whether the equation is a function, is one-to-one,onto, both, or neither. Then state whether it is discrete or continuous.  

15. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = -3x + 2 0 2 1 -1 2 -4 3 -7

-1 5 -2 8 -3 11

16. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a solid straight line. So the relation is continuous.

x y = 0.5x - 3 0 -3 1 -2.5 2 -2 3 -1.5

-1 -3.5 -2 -5 -3 -4.5

17. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the negative numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = 2x2

0 0 1 2 2 8 3 18

-1 2 -2 8 -3 18

18. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

 

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the positive numbers are left unmapped.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

x y = -5x2

0 0 1 -5 2 -20 3 -45

-1 -5 -2 -20 -3 -45

19. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . Draw a smooth curve through these points.  

   

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers};

  No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is not one-to-one because each element of the domain is not paired with a unique element of the range.   The function is not onto because the numbers less than –8 are left unmapped.   The domain has an infinite number of elements and the function can be graphed with a smooth curve. So the function is continuous.

x y = 4x2 - 8

0 -8 1 -4 2 8 3 28

-1 -4 -2 8 -3 28

20. 

SOLUTION:  To graph, substitute values for x into the equation andsolve for y . A few of the points on the graph are (0, -

1), (1, -4), (-1, 2), , (2, -25),and (-2, 23). Draw a smooth curve through these points.  

  The members of the domain are the x-values of the relation while the members of the range are the y-values. D = {all real numbers}; R = {all real numbers};   No vertical line intersects the graph in more than onepoint. So the equation is a function.   The function is both one-to-one and onto because each element of the domain is paired with a unique element of the range and each element of the range corresponds to an element of the domain.   The domain has an infinite number of elements and the relation can be graphed with a smooth curve. So the relation is continuous.

Evaluate each function. 

21. 

SOLUTION:  

Replace x with –8.  

22. 

SOLUTION:  

Replace x with 2.5.

23. DIVING The table below shows the pressure on a diver at various depths.

a. Write a relation to represent the data. b. Graph the relation. c. Identify the domain and range. Is the relation discrete or continuous? d. Is the relation a function? Explain your reasoning.

SOLUTION:  a. Let the depth measurements be the domain and the pressure be the range.{(0, 1), (20, 1.6), (40, 2.2), (60, 2.8), (80, 3.4), (100, 4)}   b. Plot each ordered pair from part a on the graph. Draw a straight line through the points. 

  c. The depth begins at 0 and can increase indefinitely. The pressure begins at 1 and can increase indefinitely so the domain and range are: D

= ; R= . The relation is

continuous because the graph represents the pressure at depths other than the given measures, such as 10 feet.   d. Each domain value is paired with only one range value. So the relation is a function.  

Find each value if

 and 

  24. 

SOLUTION:  

f (x) = 3x + 2   Replace x with –5. 

25. 

SOLUTION:  

f (x) = 3x + 2   Replace x with 9. 

26. 

SOLUTION:  

  Replace x with –3..

27. 

SOLUTION:  

  Replace x with –6. 

28. 

SOLUTION:  

  Replace x with 3.  

29. 

SOLUTION:  

  Replace x with 8 .

30. 

SOLUTION:  

 

Replace x with .

 

31. 

SOLUTION:  

 

Replace x with .

 

32. 

SOLUTION:  

 

Replace x with .

 

33. PODCASTS Chaz has a collection of 15 podcasts downloaded on his digital audio player. He decides todownload 3 more podcasts each month. The functionP(t) = 15 + 3t counts the number of podcasts P(t) he has after t months. How many podcasts will he have after 8 months?

SOLUTION:  

  Replace t with 8.  

  After 8 months Chaz will have 39 podcasts.

34. MULTIPLE REPRESENTATIONS In this problem you will investigate one-to-one and onto functions.   a. GRAPHICAL Graph each function on a separategraphing calculator screen.

  b. TABULAR Use the graphs to create a table showing the number of times a horizontal line could intersect the graph of each function. List all possibilities.   c. ANALYTICAL For a function to be one-to-one, a horizontal line on the graph of the function can intersect the function at most once. Which functions meet this condition? Which do not? Explain your reasoning.   d. ANALYTICAL For a function to be onto, every possible horizontal line on the graph of the function must intersect the function at least once. Which functions meet this condition? Which do not? Explain your reasoning.   e . GRAPHICAL Create a table showing whether each function is one-to-one and/or onto.  

SOLUTION:  

a. f (x) = x2

g(x) = 2x

h(x) = x3 - 3x2 - 5x + 6

j(x) = x3

              b.

  c. Placing a pencil on each graph so it’s parallel to the x-axis and then moving it straight up and down, only g(x) and j (x) intersect the pencil line once at a time so they are one-to-one, and f (x) and h(x) are not . d. A horizontal line on the graphs of h(x) and j (x) will intersect the graph more than once so they are onto, and f (x) and g(x) are not.   e. 

 

35. CCSS CRITIQUE Omar and Madison are finding f

(3d) for the function  Is either of them correct? Explain your reasoning.

SOLUTION:  Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct.

36. CHALLENGE Consider the functions f (x) and

 and g(a) = 33, while f (b) = 31 and g

(b) = 51. If a = 5 and b = 8, find two possible functions to represent f (x) and g(x).

SOLUTION:  Sample answer: Organize the given information into atable.   

  Analyze the information given about f (x).

  If the values of a and b are multiplied by 4, the product is one more than the value of f (a) and f (b).    Next, analyze the information given about g(x).

  If the values of a and b are multiplied by 6, the product is three less than the value of g(a) and g(b).

So the functions are: .

f (x) g(x)

f (a) = 19 g(a) = 33

f (b) = 31 g(b) = 51

a = 5, b = 8

f (x) f (a) = 19

f (5) = 19

4(5) = 20

f (x) = 4x – 1

f (a) = 4(5) – 1 = 20

f (b) = 31

f (8) = 31

4(8) = 32

f (x) = 4x – 1

f (b) = 4(8) – 1 = 31

g(x)

g(a) = 33

g(5) = 33

6(5) = 30

g(x) = 6x + 3

g(a) = 6(5) + 3 = 33

g(b) = 51

g(8) = 51

6(8) = 48

g(x) = 4x + 3

g(b) = 6(8) + 3 = 51

37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a function? Explain your reasoning.

SOLUTION:  Never; if the graph crosses the y-axis twice, then there will be two separate y-values that correspond to x = 0, which violates the vertical line test.

38. OPEN ENDED Graph a relation that can be used torepresent each of the following. a. the height of a baseball that is hit into the outfield b. the speed of a car that travels to the store, stopping at two lights along the way c. the height of a person from age 5 to age 80 d. the temperature on a typical day from 6 A.M. to 11 P.M.

SOLUTION:  a. Sample answer: let the x-axis be the time the ball is in the air and the y-axis be the height of the ball. The height of the ball is zero when the time is zero. Once the ball is thrown, the height will reach a maximum point and then decrease eventually landing on the ground.

b. Sample answer: let the x-axis be the time the car is being driven and the y-axis be the distance the car has traveled. At each stop light, time increases but distance is constant. 

c. Sample answer: let the x-axis be the age of the person and the y-axis be the height. Height increasesas a child, then more steeply, finally leveling off and remaining constant.

d. Sample answer: let the x-axis be the hours from 6 to 11 and the y-axis be the temperature. On a typical day, the morning is cool while the temperature gradually warms up to reach a maximum. The temperature gradually decreases as the sun sets.

39. REASONING Determine whether the following statement is true or false . Explain your reasoning. If a function is onto, then it must be one-to-one aswell.

SOLUTION:  Sample answer: if a function is onto then each element of the range corresponds to an element of the domain. A function that is one-to-one has each element of the domain paired to exactly one unique element of the range. The statement is false; a function is onto and not one-to-one if all of the elements of the domain correspond to an element of the range, but more than one element of the domain corresponds to the same element of the range.

40. WRITING IN MATH Explain why the vertical linetest can determine if a relation is a function.

SOLUTION:  Sample answer: A relation is a function if each x-value only pairs with one y-value. If the vertical line test fails then there is an x-value that pairs with morethan one y-value, so the relation is not a function.

41. Patricia’s swimming pool contains 19,500 gallons of water. She drains the pool at a rate of 6 gallons per minute. Which of these equations represents the number of gallons of water g, remaining in the pool after m minutes?   A  g = 19,500 – 6m   B  g = 19,500 + 6m  

C  

 

D  

SOLUTION:  Number of gallons of water in the pool = 19,500.   Patricia drains the water at a rate of 6 gallons per minute.   In m minutes, she can drain 6m gallons of water.   So the number of gallons of water remaining in the pool after m minutes is given by:

g = 19,500 – 6m   The correct choice is A.

42. SHORT RESPONSE Look at the pattern below.

If the pattern continues, what will the next term be?

SOLUTION:  

Each term of the pattern is obtained by adding to

the previous term.

Next term =

43. GEOMETRY Which set of dimensions represents atriangle similar to the triangle shown below?

F   1 unit, 2 units, 3 units G  7 units, 11 units, 12 units H  10 units, 23 units, 24 units J   20 units, 48 units, 52 units

SOLUTION:  No common ratio can be found between the dimensions of the given triangle and the sets of dimensions given in the choices F, G, H.   The dimensions of the triangle and the dimensions given in the choice J are in the ration 1:4.   So the correct choice is J.

44. ACT/SAT If  which expression is equal

to g(x + 1)? A.  1

B.  x2 + 1

C.  x2 + 2x + 1

D.  x2 – x

E.  x2 + x + 1

SOLUTION:  

  Replace x by x + 1.  

  The correct choice is C.

Solve each inequality. 

45. 

SOLUTION:  

46. 

SOLUTION:  

47. 

SOLUTION:  

  This implies:  

48. CLUBS Mr. Willis is starting a chess club at his highschool. He sent the advertisement at the right to all of the homerooms. Write an absolute value inequalityrepresenting the situation.

SOLUTION:  Let x represent the number of members.

49. SALES Ling can spend no more than $120 at the summer sale of a department store. She wants to buyshirts on sale for $15 each. Write and solve an inequality to determine the number of shirts she can buy.

SOLUTION:  Let x be the number of shirts Ling can buy.Each shirt costs $15.   So:

  Ling can buy a maximum of 8 shirts.

Solve each equation. Check your solutions. 

50. 

SOLUTION:  

  This implies:  

51. 

SOLUTION:  

  This implies:  

52. 

SOLUTION:  

  This implies:  

Simplify each expression. 

53. 

SOLUTION:  

54. 

SOLUTION:  

55. 

SOLUTION:  

Solve each equation. Check your solutions. 

56. 

SOLUTION:  

  Substitute x = 6 in the original equation. 

  So the solution is x = 6.

57. 

SOLUTION:  

  Substitute a = 4 in the original equation. 

  So the solution is a = 4.

58. 

SOLUTION:  

  Substitute x = –2 in the original equation. 

  So the solution is x = –2.

59. 

SOLUTION:  

  Substitute b = –4 in the original equation. 

  So the solution is b = –4.

60. 

SOLUTION:  

  Substitute x = 3 in the original equation. 

  So the solution is x = 3.

61. 

SOLUTION:  

  Substitute y = –4 in the original equation. 

  So the solution is y = –4.

62. 

SOLUTION:  

  Substitute c = 6 in the original equation. 

  So the solution is c = 6.

63. 

SOLUTION:  

  Substitute d = –6 in the original equation. 

  So the solution is d = –6.

64. 

SOLUTION:  

  Substitute y = 3 in the equation.  

  So the solution is y = 3.

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2-1 Relations and Functions