Name: _________________________________ NOTES: Relations

A relation is a set of ordered pairs. The domain of a relation is the set of all first coordinates of the ordered pairs (x), and the range is the set of all second coordinates ( y ).

Example 1 State the domain and range of each relation.

1. {(3, 3), (3, 4), (3, 5)} Domain = {3} Range = {3, 4, 5}

2. {(1, 2), (2, 1), (3, 2)} Domain = {1, 2, 3}Range = {1, 2}

Relations may be expressed in the form of ordered pairs, tables, graphs, and mappings.

Example 2: The relation {(1, 1), (0, 2), (3, -2)} can be expressed in each of the following ways.

The inverse of any relation is obtained by switching the coordinates in each ordered pair.

State the domain and range of each relation.

1. {(-6, 5), (-3, 8), (-6, 9), (3, 11)} 2. {(0.8, -0.8), (1.2, 0), (3.5, 4)}

3. {( 12, 1

4 ) ,(1 12,1 1

4 ),(3 12,2)}

Express the relations shown in each table, mapping, or graph as a set of ordered pairs. Then state the domain, range and inverse of the relation.

4. 5.

Draw a mapping and graph for each relation.

6. { (−2,−1 ) , (3 ,3 ) , (4,3 ) } 7. {(0, 0), (1, 1), (2, 2)}

Name: _______________________________ HOMEWORK: Relations

1

State the domain and range of each relation.

1. {(1, 1), (2, 2), (3, 1), (3, 2), (4, 1), (4, 2)}

2. {(−6 14 ,,−6 1

4 ) ,(−6 14.−1

2 ) .(5 ,3 12 ) ,(5 ,−1

2 )}

Express the relation shown in each table, mapping, or graph as a set of ordered pairs. Then state the domain and range of the relation.

3. 4. 5.

6. Cost of Admission to Water World

Draw a mapping and a graph for each relation.

7. {(0, 2), (2, 0), (2, 1), (2,2)} 8. {(-2, 1), (-2, 2), (1,1), (2,1)}

Name: _________________________________ NOTES/HOMEWORK: Functions

A special type of relation is called a function.

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Definition of FunctionA function is a relation in which each element of the domain is

paired with exactly one element of the range.

Example 1: Is {(6, -3), (4, 1), (7, -2), (-3, 1)} a function?

Since each element of the domain is paired with exactly one element of the range, the relation is a function.

The equation y=2x+1 can be written as f ( x )=2x+1. If x=3 , then f (3 )=2 (3 )+1 ,∨7. Thus, f (3), which is read f of 3 is a way of referring to the value of y that corresponds to x=3.

Example 2: If f ( x )=3 x−4 , find f (3 )∧f (−2).

f (3 )=3 (3 )−4 f (−2 )=3 (−2 )−4

¿9−4 ¿−6−4

¿5 ¿−10

Example 3: Compositions! If f ( x )=2x−1 and g ( x )=5 x+3, what is g( f (4 )). Work backwords!

Determine whether each relation is a function.

1. 2. 3.

Name: _________________________________ NOTES/HOMEWORK: Functions continued

4. {( 4 ,2 ) , (2 ,3 ) , (6 ,1 ) } 5. {(−3 ,−3 ) , (−3 ,4 ) , (−2 ,4 ) } 6. {(−1 ,0 ) , (1,0 ) }

3

7. −2 x+4 y=0 8. x2+ y2=8 9. −14x=−1

4y−2

Given f ( x )=2x−4∧g (x )=x2−4 x, find each value.

10. f (4 ) 11. g(2)

12. f (−5) 13. g(−3)

14. f (g (1 )) 15. g( f (3 ))

Name: __________________________________ CLASSWORK: Relation Machines continued

A relation machine accepts x-value numbers, performs the operation indicated by the relation, and spits out the y-value answer. Given the following relation machines, find the y-values that will be spit from the machine and determine if the relation is actually a function for the x-values given.

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Feed in these x-values Spit out the y-value answers Is it a function forthese x-values?

1. x=−3 ,−2 ,−1 ,0 ,1 ,2 ,3 y=¿

Feed in these x-values Spit out the y-value answersRound to the nearest tenth

Is it a function forthese x-values?

1. x=−2 ,−1,0 ,2 ,7 y=¿

Name: __________________________________ CLASSWORK: continued

Determine whether each relation is a function.

1. __________ 2. __________ 3. __________

5

4. __________ 5. __________ 6. __________

7. {(2 ,4 ) , (3 ,5 ) , (4 ,6 ) , (5 ,8 ) } ________ 8. {(1 ,4 ) , (2 ,3 ) , (3 ,−6 ) , (−3 ,6 ) } ________

9. {(1 ,5 ) , (1 ,7 ) , (1 ,−5 ) , (1 ,−7 ) } _______ 10. {(0 ,4 ) , (1,4 ) , (2,4 ) , (3 ,4 ) } _______

11. 2 x−3 y=4 __________ 12. xy=1 __________ 13. y=−2 ________

If f ( x )=2x−6∧g ( x )=x−2 x2, find each value.

14. f (2) 15. f ( 12 ) 16. g(−1) 17. g( f (1 ))

Name: _______________________________ Integrated Algebra Practice.

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1. Use the vertical line test to determine which graph represents a function.

2. Use the vertical line test to determine which graph represents a function.

7

Name: ___________________________________ Integrated Algebra Practice continued

3. Which graph represents a function? 4. Which graph represents a function?

8

Name: ____________________________ Integrated Algebra Practice continued

5. Which graph does not represent a function?

6. Which graph does not represent the graph of a function?

7. Which graph does not represent a function of x?

9

Name: ____________________________ Integrated Algebra Practice continued

8. Which relation represents a function?

(1) {(0 ,3 ) , (2 ,4 ) , (0 ,6 ) }

(2) {(−7 ,5), (−7 ,1) ,(−10 ,3), (−4 ,3)}

(3) {(2 ,0) ,(6 ,2) ,(6 ,−2)}

(4) {(−6 ,5 ) , (−3 ,2 ) , (1 ,2 ) , (6 ,5 ) }

9. Which set of ordered pairs represents a function?

(1) {(0 ,4 ) , (2, 4 ) , (2,5 ) }

(2) {(6 ,0 ) , (5 ,0 ) , (4 ,0 ) }.

(3) {( 4 ,1 ) , (6 ,2 ) , (6 ,3 ) , (5 ,0 ) } .

(4) {(0 ,4 ) , (1,4 ) , (0 ,5 ) , (1 ,5 ) }

10. Determine which relation is a function.

(1)

(2)

(3)

(4)

11. Determine if the relation is a function. Explain, why or why not?

12. Which relation is not a function?

(1) {(1 ,5 ) , (2 ,6 ) , (3 ,6 ) , ( 4 ,7 ) }

(2) {( 4 ,7 ) , (2,1 ) , (−3 ,6 ) , (3 ,4 ) }

(3) {(−1 ,6 ) , (1,3 ) , (2 ,5 ) , (1 ,7 ) }

(4) {(−1 ,2 ) , (0 ,5 ) , (5 ,0 ) , (2 ,−1 ) }

13. Which statement is true about the relation shown on the graph below?

(1) It is a function because there exists one x-coordinate for each y-coordinate.

(2) It is a function because there exists one y-coordinate for each x-coordiante.

(3) It is not a function because there are multiple y-values for a given x-value.

(4) It is not a function because there are multiple x-values for a given y-value.

Name: _______________________________ CLASSWORK

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For each relation, write the domain and range and tell whether the relation is a function. Justify your answer.

1. 2. 3.

4. 5. 6.

Choose the best answer. Explain your choice.

7. Given the relation { (−2,3 ) , ( x ,4 ) , (1 ,9 ) , (0 ,7 ) }, which replacement for x makes this relation a function? (1) -2 (2) 0 (3) 1 (4) 4

8. The relation {(0 ,2) , (−2 ,2 ) , (1 ,4 ) , (4 ,1 ) ,(0 ,−1)} is not a function. Which of the ordered pairs listed below, if omitted from this relation, will make the resulting set a function?

(1) (−2 ,2 ) (2) (1 ,4 ) (3) (4 ,1 ) (4) (0 ,−1 )

9. This table shows some a-values and the corresponding b-values. Which rule describes

the relationship between the variables?

(1) a=3b+1 (2) a=3b−1 (3) b=3a+1 (4) b=3a−1

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Evaluate each function for the indicated input value.

10. f (x)=−2x2−5 , find f (3) 11. g ( x )=−3|7−x|, find g(−2)

12. h ( x )=5√ x+9 , find h (0) 13. f ( x )=−4 x3+3 & g ( x )=x+2Find f (−2 )+g (3)

Name: _______________________________________ MORE NOTES ON: Evaluating Functions

Evaluate each function.12

1. h (t )=|t+2|+3 ; find h (6 ) 2. g (a )=33a−2; find g (1 )

3. w (t )=−2t+1; find w (−7 ) 4. h ( x )=4 x−2 ; find h (x+2 )

For #5 – 10, use the following functions: f ( x )=9−3 x g ( x )=5 x−7

5. f +g 6. f−g 7. g− f

8. f ∙ g 9. fg 10. gf

Name: _________________________________ HOMEWORK: Evaluating Functions

Evaluate each function

1. h (n )=−2n2+4 ; findh (4) 2. p ( t )=4 t−5 ; find p ( t−2 )

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For #3 – 8, use the following functions: f ( x )=4 x+8 g ( x )=x+3

3. f +g 4. f ∙ g 5. fg

6. f−g 7. gf 8. g− f

Name: _____________________________ NOTES: Word Problems

1. For Function 1, any y-value can be determined by using this expression: 52x+3.

The table below represents Function 2.

x -1 0 1 2 3y -1 3 7 11 15

Which statement about the two functions is true?

(1) Both functions have the same rate of change.14

(2) Function 1 has a greater rate of change than Function 2.

(3) Function 2 has a greater rate of change than Function 1.

(4) Function 1 has a positive rate of change, and Function 2 has a negative rate of change.

2. Jared is building a tree house. He pays $74 for materials and pays his friend $12 per hour to help him. If Jared spends a total of $129 on building his tree house, for how many hours did his friend work on it? _______________

Name: _____________________________ NOTES: Word Problems continued

3. The price of buying x boxes of cereal at the Corner Store is shown by the graph below. The advertisement shows the cost of buying x boxes of cereal at Bargain World.

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Determine the costs when a family buys 1, 2, 3, and 4 boxes of cereal at each store. Show your work.

Explain when a family should buy cereal at the Corner Store and when they should buy cereal at Bargain World.

4. For field trips, a museum charges a flat fee plus an additional rate for each student.

The museum uses the equation of a linear function to determine y, the total cost in dollars, if x students attend the field trip. The table below shows a partial representation of this function.

Museum Field Trip CostsNumber of Students (x)

1 2 3 4 5

Total Cost in Dollars ( y )

56 62 68 74 80

(a) What is the rate of change shown by the table and what does it represent in the problem? Show your work and explain your answer.

__________________________________________________________________________

__________________________________________________________________________

(b) The initial x-value in the table is 1. Explain why it is more appropriate for the table to start at x=1 instead of x=0.

__________________________________________________________________________

__________________________________________________________________________

5. Does the following table represent a function?

__________

Explain how you know whether the table does or does not represent a function.

__________________________________________________________________________

__________________________________________________________________________16

Name: _____________________________________ UNIT REVIEW

1. If point (d ,3) lies on the graph of 3 x− y=9, what is the value of d?

2. Which of the following is not a graph of a function?

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3. Which set of ordered pairs represents a function?

(1) {(−6 ,1 ) , (−7 ,1 ) , (1 ,−7 ) , (1 ,−6 ) }

(2) {(0 ,13 ) , (1,13 ) , (6 ,13 ) , (112 ,13 ) }

(3) {(3 ,−3 ) , (3 ,−4 ) , (−3 ,−4 ) , (−3 ,−3 ) }

(4) {(1 ,12, ), (11 ,1 ), (112 ,11) , (11 ,112 ) }

Name: ________________________________________ review continued

4. Which point does not lie on the graph of 3 x− y=9?

(1) (1 ,−6 ) (2) (2 ,3 ) (3) (3 ,0 ) (4) (0 ,−9 )

5. In which ordered pair is the x-coordinate 3 more than the y-coordinate?18

(1) (1 ,4) (2) (1 ,3) (3) (3 ,1) (4) (4 ,1)

6. Determine

(a) the domain and

(b) the range of the relation shown in the graph to the right.

(c) Is the relation a function? Explain why or why not.

7. A function is a set of ordered pairs in which no two ordered pairs have the same first element. Is y=−2x+5 a function? Explain why or why not.

Evaluate the indicated function for the given value

f ( x )=3 x−5 g ( x )=x−10

8. f (−3 ) 9. g (22 ) 10. g (−9 )

11. f (g (1 )) 12. g( f (0 ))

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Name: _______________________________ Bonus Review Sheet on Functions

(#19 Jan 16.CC) The range of the function defined as y=5x is

(1) y<0 (2) y>0 (3) y ≤0 (4) y ≥0

Given the following table:

x -2 0 4 7y 11 7 -1 -7What is the rate of change?

1. (#11 Aug 15.CC) Which representations are functions?

(1) I and II (2) II and IV (3) III, only (4) IV, only

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2. (#26 Jan 16.CC) Marcel claims that the graph represents a function. State whether Marcel is correct. Justify your answer.

3. (#14 Jan 15) Which set is a function?

(1) {(3,4), (3,5), (3,6), (3,7)} (2) {(1,2), (3,4), (4,3), (2,1)}

(3) {(6,7), (7,8), (8,9), (6,5)} (4) {(0,2), (3,4), (0,8), (5,6)}

4. (#4 Jan 12) Which graph does not represent a function?

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5. Terence has $1.00 to spend on pencils, which cost $0.29 each. Which relation shows the possible numbers of pencils he can buy and amounts of change he will receive?

(1) {(0.29, 0.71), (0.58, 0.42), (0.87, 0.13)}

(2) {(1, 0.29), (2, 0.58), (3, 0.87)}

(3) {(1, 0.71), (2, 0.42), (3, 0.13)}

(4) {(0, 0), (1, 0.29), (2, 0.58), (3, 0.87)}

6. List the domain and range, then determine if it is a function:

{(-3, 2), (1.5, 1), (1, 2), (-3, 1)}

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