Ch 1 Solution Key

Solutions KeyFoundations for Functions1

CHAPTER

ARE YOU READY? PAGE 3

1. D 2. C

3. A 4. E

5. 3 ___ 10

= 0.3 6. 3 __ 5 = 0.6

7. - 4 _ 3 = -1.

− 3 8. 5 3 _

4 = 23 _

4 = 5.75

9–12.

13. 5 _ 6 > 2 _

3 14. 3 7 _

9 < 3 10 _

12

15. -0.38 < -0.3 16. - 15 ___ 8 > -2

17. 14 ÷ 2(-3) + 1= 7(-3) + 1= -21 + 1= -20

18. 8 2 – (-12) + 15 ÷ 3= 64 + 12 + 5= 81

19. -2 (25 - 21) 2 + 11= -2 (4) 2 + 11= -2(16) + 11= -32 + 11= -21

20. 3 ( 21 - 9 ______ 6 - 1) ÷ 2

= 3 (2 - 1) ÷ 2 = 3(1) ÷ 2 = 3 ÷ 2

= 1.5 or 3 _ 2

21–24.

1-1 SETS OF NUMBERS, PAGES 6–13

CHECK IT OUT!

1a. π ≈ 3.14, 3 __ 2 = 1.5, - √ � 3 ≈ -1.73

The order is: -2, - √ � 3 , -0.321, 3 __ 2 , π

b. -2: �, �, �; - √ � 3 : �, irrational;

-0.321: �, �; 3 _ 2 : �, �;

π: �, irrational

2a. (-∞, -1] b. (-∞, 2] or (3, 11]

3a. even numbers between 1 and 9

b. {3, 4, 5, 6, 7} c. {x | x ≥ 99}

THINK AND DISCUSS

1. Possible answer: Interval notation is used to indicate infinite sets of real numbers over an interval. Roster notation is used to indicate finite or infinite sets that follow a pattern (such as multiples of 2). It is not possible to have a set represented by both methods.

2. Possible answer: No; any integer, n, can be expressed in the form n _

1 , which is a rational number.

3.

EXERCISESGUIDED PRACTICE

1. Roster notation

2. 3 √ � 2 ≈ 4.24, √ � 7 ≈ 2.6, 4 3 _ 5 = 4.6

The order is: √ � 7 , 3 √ � 2 , 4 3 _ 5 , 4.

− 6 , 5.125

√ � 7 : �, irrational; 3 √ � 2 : �, irrational;

4 3 _ 5 : �, �; 4.

− 6 : �, �;

5.125: �, �

3. - 100 ____ 4

= -25, √ � 4 = 2, 1 _

8 = 0.125, √ � 6 ≈ 2.4

The order is: - 100 ____ 4 , -6.897, 1 _

8 , √ � 4 , √ � 6

- 100 ____ 4

: �, �, �; -6.897: �, �;

1 _ 8

: �, �; √ � 4 : �, �, �, �, �;

√ � 6 : �, irrational

4. √ � 5 ≈ 2.2, π __ 2 ≈ 1.57, -

√ � 3 ≈ -1.73, -1 1 __

3 = 1.

− 3

The order is: – √ � 3 , -1 1 __ 3 , 1.

− 3 , π

__ 2

, √ � 5

– √ � 3 : �, irrational; -1 1 __ 3

: �, �;

1. −

3 : �, �; π

__ 2

: �, irrational;


5. (-10, 10] 6. (-∞, -5)

7. [1, 20) or (30, ∞) 8. one

9. {x | -5 ≤ x < 3}

10. nonnegative integer multiples of 5

11. {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

1 Holt McDougal Algebra 2

3089 Chapter 01_001-034.indd 13089 Chapter 01_001-034.indd 1 1/13/10 1:34:02 PM1/13/10 1:34:02 PM

PRACTICE AND PROBLEM SOLVING

12. 2 √ � 5 ≈ 4.47, - 4 __ 5 = -0.8,

The order is: - 4 __ 5 , -0.75, 2.33, 2 √ � 5 , 5.

− 5

- 4 __ 5 : �, �; -0.75: �, �;

2.33: �, �; 2 √ � 5 : �, irrational; 5.

− 5 : �, �

13. 1 __ 2 = 0.5, -

√ � 2 ≈ -1.4,

√ � 2 ____ 3 ≈ 0.47

The order is: -2, - √ � 2 , -1. −−

25 , √ � 2 ___ 3 , 1 __

2

-2: �, �, �; - √ � 2 : �, irrational;

-1.25: �, �; √ � 2 ____ 3 : �, irrational;

1 __ 2 : �, �

14. - √ � 9 = -3, 2π ≈ 6.28, - 7 __ 2 = -3.5

The order is: - 7 __ 2 , - √ � 9 , -1, 5.

−− 12 , 2π

- 7 __ 2 : �, �; - √ � 9 : �, �, �;

-1: �, �, �; 5. −−

12 : �, �; 2π: �, irrational

15. (-∞, 5) or (5, ∞) 16. (-15, 0)

17. [-3, 3]

18. less or equal to 3 or greater than 5 and less than or equal to 11

19. {11, 22, 33, 44, 55, 66, 77,…}

20. less than -3 or greater than 0

21. {x | -9 ≤ x ≤ -1 and x is odd}

22. Lithium, aluminum, sulfur, chlorine, calcium

23. � 24. �

25. Possible answer: Interval notation is used for ranges of numbers, but the set of atomic masses is a list of numbers.

26. negative even integers; cannot be expressed in interval notation; {x | x < 0 and x is even}

27. numbers greater than or equal to -4 and less than 8; cannot be expressed in roster notation;

{x | -4 ≤ x < 8}

28. {28, 30, 32, 34,36, 38}; cannot be expressed in interval notation; {x | 27 < x < 39 and x is even}

29. numbers greater than 0 and less than 1; cannot be expressed in roster notation; (0, 1)

30. (-∞, -4) or (4, ∞);{x | x < -4 or x > 4}

31. (-∞, 2) or (2, ∞);{x | x ≠ 2}

32. (∞, 2] or (3, 5);{x | x ≤ 2 or 3 < x < 5}

33. (1, 10); {x | 1 < x < 10}

34. (-∞, 6) or (10, ∞);{x | x < 6 or x > 10}

35. (-∞, 5) or (5, 10];{x | x < 5 or 5 < x ≤ 10}

36. true

37. False; possible answer: 3 is a real number but not irrational.

38. False; possible answer: -4 is an integer but not a whole number.

39. true

40. Soccer ball A: C = 2πr = 2π (4.36) ≈ 27.38 in. Therefore, the size of ball A is 5. Soccer ball B: C = πd = (7.54) π ≈ 23.68 in. Therefore, the size of ball B is 3.

Soccer ball C: V = 4 __ 3 π r 3

276.2 = 4 __ 3 π r 3

( 3 ___ 4π

) (276.2) = ( 3 ___ 4π

) ( 4π

___ 3 ) r 3

65.94 ≈ r 3

3 √ �� 65.94 ≈

3

√ � r 3 4.04 ≈ r So, C = 2πr = 2π (4.04) ≈ 25.38 in. Therefore, the size of ball C is 4.

41. size 3: {x | 11 ≤ x ≤ 12} size 4: {x | 12 ≤ x ≤ 13} size 5: {x | 14 ≤ x ≤ 16}

42. size 3: (0, 8) size 4: [8, 12] size 5: (12, ∞)

43. The circumference is always irrational because it is the product of an irrational number, π, and a rational number, the diameter d.

44a. �, �

b. 97 ____ 186

≈ 0.522, 117 ____ 310

≈ 0.377

The order is: Moon, Venus, Mercury, Mars.

c. The round-trip to Venus would take longer because twice the average distance between Earth and Venus is about 0.555 AU and the average distance between Earth and Mars is about 0.522 AU.

45. (∞, -1] or (3, 6) or [9, ∞)

46.

47.

48.

49.

50.



51.

52a. {talc, gypsum, calcite, fluorite, apatite}

b. 5; orthoclase to diamond are harder

c. Neither; quartz is harder than window glass but apatite is softer than window glass.

53. � 54. �

55. �

56. No; possible answer: square roots of some numbers are not irrational. √ � 9 = 3, and 3 is rational.

57a. interior designer, police officer, pediatric nurse, marine biologist, astronaut

b. The order would not change, because each salary is increased by the same amount.

c. The order would not change, because the amount by which each salary is increased is relative to its original amount, and that does not allow one salary to increase so much or so little as to change the order.

d. {46,000, 52,900, 59,800, 79,350, 106,950}

58. Possible answer: rational: 5 __ 2

; irrational: √ � 2 ___ 2 ; 5 is in

the set

59. Possible answer: rational: 6; irrational: 6 √ � 3 ; 5 is in the set.

60. Possible answer: rational: 11: irrational: 4 √ � 5 ; 5 is in the set.

61. Possible answer: rational: 3 3 __ 2 ; irrational: π; 5 is not

in the set.

62. Possible answer: Mathematical and everyday sets are similar because they are both made up of elements. They are different because mathematical sets can be infinite.

TEST PREP

63. D 2(-2) = -4

64. F

3 __ 7 ≈ 0.42,

√ � 3 ___

2 ≈ 0.866

65. B - √ � 4 = -2, - 5 __

3 = 1.

− 6 , 1 1 __

2 = 1.5

66. J

CHALLENGE AND EXTEND

67. finite; � 68. infinite; �

69. finite; �, �, �, � 70. infinite; �

71a. Possible answer: 3.141

b. Possible answer: 3.142

SPIRAL REVIEW

72. Possible answer: −−

AB and −−

BC

73. Possible answer: AEGD and BFHC

74. Possible answer: AEGD and ABFE

75. 1.065(21.49 + 11.59 + 12.95)= 1.065(46.03)= 49.02The cost is $49.02.Since she could only have 3 bills, then the only combination Debra could have is $20, $20, $10.

76. 3.7 ___ s = 1 ____ 120

s = 3.7(120) s = 444 cm The square has area of (444) 2 = 197,136 cm 2 .

1-2 PROPERTIES OF REAL NUMBERS, PAGES 14–19

CHECK IT OUT!

1a. additive inverse: -500; multiplicative inverse: 1 _ 500

b. additive inverse: 0.01; since -0.01 = - 1 ____ 100

,

multiplicative inverse: -100

2a. Comm. Prop. of Mult. b. Assoc. Prop. of Mult.

3. 20% = 2(10%) [2(10%)]15.60 = 2[(10%)15.60] = 2[(0.1)(15.60)] = 2(1.56) = $3.12

4a. always true by the Additive Inverse Property

b. sometimes true;true when a = 0, b = 1, c = 2false when a = 1, b = 2, c = 3

THINK AND DISCUSS

1. Possible answer: No, the Commutative Property does not apply to subtraction or division because the order in subtraction and division is essential.

2. Possible answer: The product of 0 and another number is always 0, so you cannot multiply 0 by any number and get a product of 1.

3.


1. additive inverse: 36; multiplicative inverse: - 1 _ 36

2. additive inverse: 0.05; since -0.05 = 1 ____ -20

,

multiplicative inverse: -20

3. additive inverse: -2 √ � 2 ; multiplicative inverse: 1 _ 2 √ � 2



4. additive inverse: - 2 _ 5 ; multiplicative inverse: 5 _

2

5. additive inverse: 1 _ 500

; multiplicative inverse: -500

6. additive inverse: -0.25; since 0.25 = 1 _ 4 ,

multiplicative inverse: 4

7. Assoc. Prop. of Mult. 8. Comm. Prop. of Add.

9. Comm. Prop. of Mult.

10. 3(2.55) = 3(2 + 0.5 + 0.05) = 3(2) + 3(0.5) + 3(0.05) = 6 + 1.5 + 0.15 = $7.65

11. 33 1 __ 3 % = 1 __

3

(33 1 __ 3 %) (21.99)

= ( 1 __ 3 ) (21 + 0.99)

= ( 1 __ 3 ) 21 + ( 1 __

3 ) 0.99

= 7 + 0.33 = $7.33

12. always true by the Distributive Property

13. sometimes true; true when a = b; false when a = 1 and b = 2

14. sometimes true; true when a = 0; false when a = 1, b = 2, and c = 3


15. additive inverse: 2.5; since -2.5 = - 5 __ 2 ,

multiplicative inverse: - 2 __ 5

16. additive inverse: -0.75; since 0.75 = 3 __ 4 ,

multiplicative inverse: 4 __ 3

17. additive inverse: -2π; multiplicative inverse: 1 ___ 2π

18. additive inverse: 2 __ 3 ; multiplicative inverse: - 3 __

2

19. additive inverse: - 1 ___ 20

; multiplicative inverse: 20

20. additive inverse: -6231; multiplicative inverse: 1 _____ 6231

21. Distributive Property 22. Comm. Prop. of Mult.

23. Additive Identity Property

24. 9%(150) = 0.09(100 + 50) = 0.09(100) + 0.09(50) = 9 + 4.5 = $13.50

25. 5(2.00) = 10.00 5(0.04) = 0.2 10.00 - 0.20 = $9.80

26. always true by the Distributive Property

27. never true by the Multiplicative Inverse Property

28. 2(8.88) + 3(14.99) = 2(8 + 0.88) + 3(14 + 0.99) = 2(8) + 2(0.88) + 3(14) + 3(0.99) = 16 + 1.76 + 42 + 2.97 = $62.73

29. 4(11.99) - 2(8.88) = 4(11 + 0.99) - 2(8 + 0.88) = 4(11) + 4(0.99) - 2(8) - 2(0.88) = 44 + 3.96 - 16 - 1.76 = $30.20

30. 4(0.85)(14.99) = 3.4(15 - 0.01) = 51 - 0.034 = 50.966 ≈ $50.97

31. 3(0.9)(9.96) + 5(0.75)(11.99) = 2.7(9.96) + 3.75(11.99) = (2 + 0.7)(9 + 0.96) + (3 + 0.75)(11 + 0.99) = 71.8545 ≈ $71.85

32. time = distance _______ speed

= 11 + 32 + 38 ___________ 40

= 81 ___ 40

≈ 2 h

33. miles that can be driven on one tank: 24 × 8 = 192 mi length of one loop: 11 + 32 + 38 = 81 mi

number of loops per tank: miles per tank

____________ length of loop

= 192 ____ 81

≈ 2 loops

34. length of new loop: 81 × 1.2 = 97.2 mi

number of miles driven in 10 h: x mi ____ 10 h

= 40 mi _____ 1 h

x = 400 mi

loops in 10 h day: miles in 10 h ___________ length of loop

= 400 ____ 97.2

≈ 4 loops

35. (10 + 5) + 23 = 10 + (5 + 23); Assoc. Prop. of Add.

36. 12 + 11 _ 15

x = 11 ___ 15

x + 12; Comm. Prop. of Add.

37. j + 0 = j; Additive Identity Property

38. 5 · 4 + 5 · 3 = 5 · (4 + 3); Distributive Property

39. 4 _ 5

· 5 __ 4

= 1; Multiplicative Inverse Prop.

40. ab = ba; Comm. Prop. of Mult.

41. Yes; by Distributive Property, both methods give the same results.

42. Find the ticket price, which is 60% of $185: 10% of 185 = 0.1(185) = 18.5 60% of 185 = 6(18.5) = $111.00 Add $16 + $12 = $28 for fees and surcharge: $111 + $28 = $139



43. Possible answer: The set of integers is made up of the set of natural numbers, their additive inverses, and the additive identity. The set of rational numbers is made up of the set of numbers that can be expressed as a ratio of two natural numbers, their additive inverses, and the additive identity.

44. Assoc. Prop. of Add.; Additive Inverse Prop.

45. Multiplicative Identity Prop.

46. Comm. Prop. of Add; Comm. Prop. of Mult.

47. Distrib. Prop.; Assoc. Prop. of Add.

48. Distrib. Prop.; Comm. Prop. of Add.

49. Distrib. Prop.

50a. (18 + 13) – 24 = 7

b. yes; possible answer: 9 + 19 = 4 mod 24, 19 + 9 = 4 mod 24

c. yes; possible answer: 5 + (12 + 20) = 13 mod 24, (5 + 12) + 20 = 13 mod 24

51. Possible answer: By the Distributive Property, the savings is 5% not 10%.5% (labor) + 5% (parts) = 5% (labor + parts) = 5% (total)

52. Possible answer: Opposites are used for addition. A number and its opposite have different signs. Reciprocals are used for multiplication. A number and its reciprocal have the same sign.

TEST PREP

53. D 54. J

55. C

56. 4(1 + 3) = 4(4) = 16 by the order of operations; 4(1 + 3) = 4(1) + 4(3) = 4 + 12 = 16 by the Distributive Property.


57. n = 4 ( 1 __ n )

(n)n = 4 ( 1 __ n ) (n)

n 2 = 4 n = ± √ � 4 = ±2 n = 2 since n is positive

58a. 3 + 5 = 8, 5 + 3 = 8; 3 - 5 = -2, 5 - 3 = 2; 3 · 5 = 15, 5 · 3 = 15; 3 ÷ 5 = 0.6, 5 ÷ 3 = 1.

− 6 ;

the pair 3 + 5 and 5 + 3, and the pair 3 · 5 and 5 · 3

b. a + b and b + a; a · b and b · a always represent natural numbers.

c. Natural numbers are closed under addition and multiplication.

d. Integers are closed under addition, subtraction, and multiplication.

SPIRAL REVIEW

59. Area of garden last summer: 12 × 8 = 96 ft 2 Area of garden this summer: 16 × 10 = 160 ft 2 Let n represent the percent increase. 96 + 96n = 160 __________ - 96 ____ - 96 96n = 64

96n ____ 96

= 64 ___ 96

n = 0.66 −

6 n ≈ 66.7%

60. π (≈ 3.14) 61. -4 √ � 2 (≈ -5.66)

62. -4 √ � 2 and π 63. (-10, 0]

64. {x | -10 < x ≤ 0} 65. cannot be notated

1-3 SQUARE ROOTS, PAGES 21–26

CHECK IT OUT!

1. - √ �� 64 < - √ �� 55 < - √ �� 49 -8 < - √ �� 55 < -7 7.4 2 = 54.76 7.5 2 = 56.25- √ �� 55 ≈ -7.4

2a. √ �� 48 = √ �� 16 · 3 = 4

√ � 3

b. √ ��

36 ___ 16

= 6 __ 4

= 3 __ 2

c. √ � 5 · √ �� 20 = √ �� 5 · 20 = √ �� 100 = 10

d. √ �� 147 _____ √ � 3

= √ ��

147 ____ 3

= √ �� 49 = 7

3a. 3 √ � 5 ____

√ � 7

= 3 √ � 5

____ √ � 7

· √ � 7 ___ √ � 7

= 3 √ �� 35

_____ 7

b. 5 ____ √ �� 10

= 5 ____ √ �� 10

· √ �� 10

____ √ �� 10

= 5 √ �� 10

_____ 10

= √ �� 10

____ 2

4a. 3 √ � 5 + 10 √ � 5 = (3 + 10) √ � 5 = 13 √ � 5

b. √ �� 80 - 5 √ � 5 = √ �� 16 · 5 - 5 √ � 5 = 4 √ � 5 - 5 √ � 5 = (4 - 5) √ � 5 = - √ � 5

THINK AND DISCUSS

1. Both are equal to 15 √ � 2 . 3 √ �� 50 = 3 √ �� 25 · 2 = 15 √ � 2 5 √ �� 18 = 5 √ �� 9 · 2 = 15 √ � 2

2. √ �� 16 · √ � 4 = 4 · 2 = 8 or √ �� 16 · √ � 4 = √ �� 64 = 8



3.


1. radicand

2. √ �� 64 < √ �� 75 < √ �� 81 8 < √ �� 75 < 9 8.6 2 = 73.96 8.7 2 = 75.69 √ �� 75 ≈ 8.7

3. √ �� 16 < √ �� 20 < √ �� 25 4 < √ �� 20 < 5 4.4 2 = 19.36 4.5 2 = 20.25 √ �� 20 ≈ 4.5

4. - √ �� 100 < - √ �� 93 < - √ �� 81 -10 < - √ �� 93 < -9 9.6 2 = 92.16 9.7 2 = 94.09 - √ �� 93 ≈ -9.6

5. √ � 9 < √ �� 13 < √ �� 16 3 < √ �� 13 < 4 3.6 2 = 12.96 3.7 2 = 13.69 √ �� 13 ≈ 3.6

6. - √ �� 300 = - √ �� 100 · 3 = -10 √ � 3

7. √ �� 24 · √ � 6 = √ �� 144 = 12

8. √ �� 72 ____ √ � 2

= √ �� 36 = 6

9. √ �� 80 = √ �� 16 · 5 = 4 √ � 5

10. 1 ___ √ � 2

= 1 ___ √ � 2

· √ � 2 ___ √ � 2

= √ � 2 ___ 2

11. 5 √ � 6 _____

- √ � 3

= 5 √ � 6

_____ - √ � 3

· - √ � 3

_____ - √ � 3

= -5 √ �� 18

_______ 3

= -5 √ �� 9 · 2

________ 3

-5(3) √ � 2

________ 3

= -5 √ � 2

12. √ �� 50

____ √ �� 12

= √ �� 50

____ √ �� 12

· √ �� 12 ____ √ �� 12

=

√ �� 600 ______

12

= √ �� 100 · 6

________ 12

= 10 √ � 6

_____ 12

= 5 √ � 6

____ 6

13. √ � 3 ______

- √ �� 21

= √ � 3 ______

- √ �� 21 · - √ �� 21 ______

- √ �� 21

= -

√ �� 63 ______

21

= - √ �� 9 · 7

_______ 21

= -3 √ � 7 ______ 21

= - √ � 7

___ 7

14. 6 √ � 7 + 7 √ � 7 = (6 + 7) √ � 7 = 13 √ � 7

15. 5 √ �� 32 - 15 √ � 2 = 5 √ �� 16 · 2 - 15 √ � 2 = 5 · 4 √ � 2 - 15 √ � 2 = 20 √ � 2 - 15 √ � 2 = (20 - 15) √ � 2 = 5 √ � 2

16. 4 √ � 5 + √ �� 245 = 4 √ � 5 + √ �� 49 · 5 = 4 √ � 5 + 7 √ � 5 = (4 + 7) √ � 5 = 11 √ � 5

17. - √ �� 50 + 6 √ � 2 = - √ �� 25 · 2 + 6 √ � 2 = -5 √ � 2 + 6 √ � 2 = (-5 + 6) √ � 2 = √ � 2


18. √ �� 49 < √ �� 60 < √ �� 64 7 < √ �� 60 < 8 7.7 2 = 59.29 7.8 2 = 60.84 √ �� 60 ≈ 7.7

19. - √ �� 16 < - √ �� 15 < - √ � 9 -4 < - √ �� 15 < -3

3.8 2 = 14.44 2.9 2 = 15.21

- √ �� 15 ≈ -3.9

20. √ �� 36 < √ �� 47 < √ �� 49 6 < √ �� 47 < 7 6.8 2 = 46.24 6.9 2 = 47.61 √ �� 47 ≈ 6.9

21. √ �� 81 < √ �� 99 < √ �� 100 9 < √ �� 99 < 10 9.8 2 = 96.04 9.9 2 = 98.01 √ �� 99 ≈ 9.9

22. √ �� 162 = √ �� 81 · 2 = 9 √ � 2

23. - √ ��

1 ____ 121

= - 1 ___ 11

24. √ ��

50 ___ 9

= √ �� 25 · 2

_______ 3

= 5 √ � 2 ____ 3

25. -2 √ �� 10 ·

√ � 8

= -2 √ �� 80 = -2 √ �� 16 · 5 = -2(4) √ � 5 = -8 √ � 5

26. √ �� 288

_____ √ � 8

= √ �� 144 · 2 ________ √ �� 4 · 2

= 12 √ � 2 _____ 2 √ � 2

= 6

27. √ �� 85 · √ � 5 = √ �� 425 = √ �� 25 · 17 = 5 √ �� 17



28. 2 √ �� 126 ______

√ �� 14

= 2 √ �� 9 · 14

________ √ �� 14

= 2(3) √ �� 14

_______ √ �� 14

= 6

29. - √ �� 189 = - √ �� 9 · 21 = -3 √ �� 21

30. 2 ___ √ � 3

= 2 ___ √ � 3

· √ � 3

___ √ � 3

= 2 √ � 3

____ 3

31. 3 √ �� 27 ______

2 √ � 6

= 3 √ �� 27 _____ 2 √ � 6

·

√ � 6 ____

√ � 6

= 3

√ �� 162 _______

2 · 6

=

√ �� 81 · 2

_______ 2 · 2

= 9 √ � 2 ____ 4

32. - 18 ____

√ � 6

= - 18 ____

√ � 6

· √ � 6

___ √ � 6

= - 18

√ � 6 ______

6

= -3 √ � 6

33. √ � 11 ______ 5 √ �� 132

= √ � 11 ______ 5 √ �� 132

· √ �� 132

_____ √ �� 132

= √ �� 1452

______ 5(132)

= √ �� 484 · 3

________ 660

= 22 √ � 3

______ 22 · 30

= √ � 3

___ 30

34. 4 √ � 3 - 9 √ � 3 = (4 - 9) √ � 3 = -5 √ � 3

35. √ �� 112 + √ �� 63 = √ �� 16 · 7 + √ �� 9 · 7 = 4 √ � 7 + 3 √ � 7 = 7 √ � 7

36. √ � 8 - 15 √ � 2 = √ �� 4 · 2 - 15 √ � 2 = 2 √ � 2 - 15 √ � 2 = -13 √ � 2

37. √ �� 12 + 7 √ �� 27 = √ �� 4 · 3 + 7 √ �� 9 · 3 = 2 √ � 3 + 7 · 3 √ � 3 = 2 √ � 3 + 21 √ � 3 = 23 √ � 3

38. √ �� 45 + √ �� 20 = √ �� 9 · 5 + √ �� 4 · 5 = 3 √ � 5 + 2 √ � 5 = 5 √ � 5

39. 5 √ �� 28 - 2 √ � 7 = 5 √ �� 4 · 7 - 2 √ � 7 = 5 · 2 √ � 7 - 2 √ � 7 = 10 √ � 7 - 2 √ � 7 = 8 √ � 7

40. 2 √ �� 48 + 2 √ �� 12 = 2 √ �� 16 · 3 + 2 √ �� 4 · 3 = 2 · 4 √ � 3 + 2 · 2 √ � 3 = 8 √ � 3 + 4 √ � 3 = 12 √ � 3

41. √ �� 150 - 8 √ � 6 = √ �� 25 · 6 - 8 √ � 6 = 5 √ � 6 - 8 √ � 6 = -3 √ � 6

42. Let x represent the size of side of the square. x · x = 4000 x 2 = 4000

√ � x 2 = √ �� 4000 x ≈ 63.25 The dimensions would be about 63.25 m by 63.25 m.

43. perimeter = 12 √ ��

40 ___ 5

= 12 √ � 8

≈ 12(2.8284) ≈ 33.9 cm

44. perimeter = 14 √ ��

90 ___ 6

= 14 √ �� 15

≈ 14(3.873) ≈ 54.2 ft

45. perimeter = 14 √ ��

300 ____ 6

= 14 √ �� 50

≈ 14(7.07) ≈ 99.0 in.

46. length of diagonal = √ � 2 · length of side

= √ � 2 ·

√ �� 8100

= √ �� 16,200 ≈ 127.28 ft

47. The dimensions are about 50 in. by 50 in.; four canvases would cover a total of 2400 in 2 , which is approximately 2500 in 2 ; √ �� 2500 = 50, so the sides are about 50 in.

48. √ �� 900

_____ √ �� 20

= √ �� 45 = √ �� 9 · 5 = 3 √ � 5

49. 3 √ �� 50 · 3 √ � 8 = 9 √ �� 400 = 9(20)= 180

50. -3 √ � x + √ � 9x = -3 √ � x + √ �� 3 · 3 · x = -3 √ � x + 3 √ � x = 0

51. 2 √ � 5 - 5 √ � 2 cannot simplify further

52. √ �� 25x - 6 √ � x = √ �� 5 · 5 · x - 6 √ � x = 5 √ � x - 6 √ � x = -1 √ � x , or - √ � x

53. 3 √ � 7 + 1 ________ √ � 5

= 3 √ � 7 + 1 ________ √ � 5

· √ � 5

___ √ � 5

= √ � 5 (3 √ � 7 + 1)

____________ 5

= 3 √ �� 35 + √ � 5

__________ 5



54. 4 √ �� 10 - √ �� 90 ___________

√ � 2

= 4 √ �� 10 - √ �� 90

___________ √ � 2

· √ � 2 ____

√ � 2

= √ � 2 (4 √ �� 10 - √ �� 90 )

_______________ 2

= 4 √ �� 20 - √ �� 180

_____________ 2

= 4

√ �� 4 · 5 -

√ �� 36 · 5 _________________

2

= 4 · 2 √ � 5 - 6 √ � 5

_____________ 2

= 8 √ � 5 - 6 √ � 5

__________ 2

= 2 √ � 5

____ 2

= √ � 5

55. 4 √ �� 32 _____

√ � 5

= 4 √ �� 32

_____ √ � 5

· √ � 5

___ √ � 5

= 4

√ �� 160 _______

5

= 4 √ �� 16 · 10

_________ 5

= 4 · 4 √ �� 10

________ 5

= 16 √ �� 10

______ 5

56. √ �� 75x + √ �� 45x = √ �� 5 · 5 · 3 · x + √ �� 5 · 3 · 3 · x = 5 √ � 3x + 3 √ � 5x

57. side length of a ward = √ ��

8,640,000

_________ 24

= √ �� 360,000 = 600 ft The dimensions of a ward are 600 ft by 600 ft.

58. 10 acres = 10(43,560) = 435,600 ft 2 side length = √ �� 435,600 = 660 ft

59. 2 mi 2 = 2(27,880,000) = 55,760,000 ft 2 side length = √ �� 55,760,000 ≈ 7467.3 ft

60. 5 hectare = 5(107,600) = 538,000 ft 2 side length = √ �� 538,000 ≈ 733.5 ft

61. 6.2 km 2 = 6.2(10,760,000) = 66,712,000 ft 2 side length = √ �� 66,712,000 ≈ 8167.7 ft

62. sometimes true; possible answer: true when both a and b equal 4, false when a equals 5 and b equals 3

63. always true; possible answer: when a = 4 and b = 9;

√ �� 4(9)

_____ √ � 4

= √ �� 36

____ √ � 4

= 6 __ 2 = 3 = √ � 9

64. sometimes true; possible answer: true for a = 2 and b = 1, false for a = 2 and b = 4

65. no; for example, √ �� 3 + 3 = √ � 6 ≠ 3

66. Possible answer: no; √ � 2 is an irrational number but is rounded by the calculator. The square of the rounded number does not equal 2.

67a. t = √ ��

h ____ 0.82

t = √ ��

50 ____ 0.82

t ≈ 7.81 s

b. t = √ ��

h ____ 4.89

t = √ ��

50 ____ 4.89

t ≈ 3.20 s

TEST PREP

68. A A) √ �� 20 = 2 √ � 5

B) √ � 8 · √ � 5 = 2 √ �� 10 C) 2 √ �� 10

D) 5 √ � 8

____ √ � 5

= 5 √ � 8 · √ � 5

_________ 5

= 2 √ �� 10

69. Hperimeter: 4 √ �� 30 ≈ 4(5.48) ≈ 22 m

70. D

1 ___ √ � 2

, 1, √ � 2 , 2

0.707, 1, 1.414, 2

71. −−

AC =

√ �� −−

AB 2 +

−− BC 2

−−

AC = √ �� 8 2 + 4 2

−− AC = √ �� 64 + 16

−−

AC = √ �� 80

−− AC ≈ 8.9 ft


72. a √ � b - 3a √ �� 5ab _____________

3 √ � b

= 5 √ � 6 - 3(5) √ �� 5 · 5 · 6

__________________ 3 √ � 6

= 5 √ � 6 - 15 · 5 √ � 6

______________ 3 √ � 6

= 5 - 75 ______ 3

= - 70 ___ 3

73a. Small right triangle: a 2 + b 2 = c 2 a 2 + 6 2 = (6 √ � 2 ) 2 a 2 + 36 = (6 √ � 2 )(6 √ � 2 ) a 2 + 36 = (36 · 2) a 2 + 36 = 72 _______ - 36 ____ - 36 a 2 = 36

√ � a 2 = √ �� 36 a = 6 in.

Large right triangle:

a 2 + b 2 = c 2

6 2 + 12 2 = c 2

36 + 144 = c 2

180 = c 2

√ �� 180 = √ � c 2

√ �� 36 · 5 = c 6 √ � 5 in. = c

b. A = (b × h)

______ 2

= 18 × 6 ______ 2

= 108 ____ 2

= 54 in 2

c. perimeter: 18 + 6 √ � 2 + 6 √ � 5 in.



74. √ �� x 3 y 5

________ x 2 √ �� 48 y 3

= √ �� x 2 · x · y 4 · y

_______________ x 2 √ �� 16 · 3 · y 2 · y

= x y 2 √ � xy

________ 4 x 2 y √ � 3y

= y √ � xy

______ 4x √ � 3y

= y √ � xy

______ 4x √ � 3y

· √ � 3y

____ √ � 3y

= y √ �� 3x y 2

_______ 4x · 3y

= y · y √ � 3x

________ 4x · 3y

= y √ � 3x

_____ 12x

SPIRAL REVIEW, PAGE 26

75. tetrahedron or triangular pyramid

76. cylinder 77. triangular prism

78. -7 < x ≤ 1 79. 1.5 < x < 8

80. 2 ≤ x ≤ 12 81. 3 __ 4 < x < 5 __

2

82. Identity Property of Multiplication

83. Commutative Property of Addition

84. Assoc. Prop. of Mult.

85. Distributive Prop.

1-4 SIMPLIFYING ALGEBRAIC EXPRESSIONS, PAGES 27–32

CHECK IT OUT!

1a. age = 18 + y

b. 60 s in 1 min; 60 min in 1 h60 × 60 s in 1 hseconds = 3600h

2. x 2 y - x y 2 + 3y= (2) 2 (5) - (2)( 5) 2 + 3(5)= 4(5) - 2(25) + 15= 20 - 50 + 15= -15

3. -3(2x - xy + 3y) - 11xy= -6x + 3xy - 9y - 11xy= -6x - 8xy - 9y

4a. Let h represent the number of Hawaii packages sold agent will make = 50h + 80(100 - h)

= 50h + 8000 - 80h = -30h + 8000

b. agent will make = -30h + 8000 = -30(28) + 8000 = -840 + 8000 = $7160

THINK AND DISCUSS

1. Possible answer: An expression with 5 terms will have 4 addition or subtraction symbols, 1 separating each pair of terms.

2. Possible answer: When you add like terms such as 3x + 4x, you can use the Distributive Property to rewrite the sum. In this case, the sum becomes

(3 + 4)x. This expression can be simplified to 7x.

3.


1. cost = 0.79c 2. area = 8�

3. a 2 + b 2 - 2ab= ( 5) 2 + (8) 2 - 2(5)(8)= 25 + 64 - 80= 9

4. 3xy __________

x 2 - 9y + 2

= 3(2)(4) _____________

( 2) 2 - 9(4) + 2

= 24 __________ 4 - 36 + 2

= 24 ____ -30

= - 4 __ 5

5. -8a + 9 - 5a + a= -12a + 9

6. -2(2x + y) - 7x + 2y= -4x - 2y - 7x + 2y= -11x

7. 1 + (ab - 5a)5 - b 2 = 1 + 5ab - 25a - b 2

8a. total calories used = 9(r) + 7(60 - r) = 9r + 420 - 7r = 2r + 420

b. calories used = 2r + 420 = 2(20) + 420 = 40 + 420 = 460 Calories


9. If angle measures x°, supplementry angle = (180 - x)°

10. bagels = d ____ 0.60

11. 6c - 3 c 2 + d 3 = 6(5) - 3 (5) 2 + 3 3 = 30 - 3(25) + 27= 30 - 75 + 27= -18

12. y 2 - 2x y 2 - x= (3) 2 - 2(2)( 3) 2 - 2= 9 - (4)(9) - 2= 9 - 36 -2= -29

13. 3 a 2 b - a b 3 + 5= 3( 5) 2 (2) - (5) (2) 3 + 5= 3(25)(2) - 5(8) + 5= 150 - 40 + 5= 115



14. 2s - t 2 ______ s t 2

= 2(5) - ( 3) 2

_________ (5)( 3) 2

= 10 - 9 ______ 5(9)

= 1 ___ 45

15. -x - 3y + 4x - 9y + 2= 3x - 12y + 2

16. -4(-a + 3b) - 3(a - 5b)= 4a - 12b - 3a + 15b= a + 3b

17. 5 - (3m + 2n)= 5 - 3m - 2n

18. x(4 + y) - 2x(y + 7)= 4x + xy -2xy - 14x= -10x -xy

19a. Let m represent the number of muffins. total time = 30(m) + 50(10 - m) = 30m + 500 - 50m = -20m + 500

b. time = -20m + 500 = -20(2) + 500 = -40 + 500 = 460 min = 7 h 40 min

20. -a( a 2 + 2a - 1)= - a 3 - 2 a 2 + a= - (2) 3 - 2( 2) 2 + 2= -(8) - 2(4) +2= -8 - 8 + 2= -14

21. ( 2g - 1) 2 - 2g + g 2 = (2g - 1)(2g - 1) - 2g + g

2

= 4 g 2 - 2g - 2g + 1 - 2g + g

2

= 5 g 2 - 6g + 1= 5( 3) 2 - 6(3) + 1= 5(9) - 18 + 1= 45 - 18 + 1= 28

22. u 2 - v 2 _______ uv

= u 2 ___ uv - v 2 ___ uv

= u __ v - v __ u

= 4 __ 2 - 2 __

4

= 2 - 1 __ 2

= 3 __ 2

23. a 2 - 2 ( b 2 - a)

_____________ 2 + a

= a 2 - 2 b 2 + 2a ____________ 2 + a

= ( 3) 2 - 2(5 ) 2 + 2(3)

________________ 2 + 3

= 9 - 2(25) + 6

____________ 5

= 9 - 50 + 6 __________ 5

= - 35 ___ 5

= -7

24. x (x + 3 ) 2 x 2 + 9 x 2 + 6x + 9

1 (1 + 3) 2 1 2 + 9 1 2 + 6(1) + 9

= (4) 2 = 1 + 9 = 1 + 6 + 9

= 16 = 10 = 16

2 (2 + 3) 2 2 2 + 9 2 2 + 6(2) + 9

= ( 5) 2 = 4 + 9 = 4 + 12 + 9

= 25 = 13 = 25

3 (3 + 3) 2 3 2 + 9 3 2 + 6(3) + 9

= ( 6) 2 = 9 + 9 = 9 + 18 + 9

= 36 = 18 = 36

4 (4 + 3) 2 4 2 + 9 4 2 + 6(4) + 9

= ( 7) 2 = 16 + 9 = 16 + 24 + 9

= 49 = 25 = 49

Therefore (x + 3) 2 = x 2 + 6x + 9.

25. x (x - 4) 2 x 2 + 16 x 2 - 8x + 16

1 (1 - 4) 2 1 2 + 16 1 2 - 8(1) + 16

= (- 3) 2 = 1 + 16 = 1 - 8 + 16

= 9 = 17 = 9

2 (2 - 4) 2 2 2 + 16 2 2 - 8(2) + 16

= (-2) 2 = 4 + 16 = 4 - 16 + 16

= 4 = 20 = 4

3 (3 - 4) 2 3 2 + 16 3 2 - 8(3) + 16

= (-1 ) 2 = 9 + 16 = 9 - 24 + 16

= 1 = 25 = 1

4 (4 - 4) 2 4 2 + 16 4 2 - 8(4) + 16

= 0 = 16 + 16 = 16 - 32 + 16

= 32 = 0

Therefore (x - 4) 2 = x 2 - 8x + 16.

26a. Let m represent an m-minute commercial first Super Bowl = 85,000m Super Bowl XXXVIII = (2,300,000)(2)m = 4,600,000m

b. length of commercial: 85,000m = 170,000

85,000m

________ 85,000

= 170,000

_______ 85,000

m = 2 cost during Super Bowl XXXVIII: cost = 4,600,000(2) = $9,200,000 Super Bowl XXXVIII costs about 54 times as much.



c. first Super Bowl: 60,000,000 viewers

cost per 1000 viewers = $85,000

_______ 60,000

m ≈ 1.42m

Super Bowl XXXVIII: 800,000,000 viewers

cost per 1000 viewers = $4,600,000

__________ 800,000

m ≈ 5.75m

d. first Super Bowl cost = 1.42(2) = $2.84 Super Bowl XXXVIII cost = 5.75(2) = $11.50 Super Bowl XXXVIII costs about 4 times as much

per 1000 viewers.

27. 2a + a + 2b + (a + b) + a + (2a + b)= 3a + 2b + a + b + a + 2a + b= 7a + 4b

28. (2 x 2 - 5) + (9 - x) + (x + 2) + (3x - 6)

+ ( x 2 - 4) + (x + 3) + x + x= 2 x 2 - 5 + 9 - x + x + 2 + 3x - 6 + x 2 - 4 + x + 3 + 2x= 3 x 2 + 6x - 1

29a. total budgeted cost = 100d + 275(15 - d) = 100d + 4125 - 275d = -175d + 4125

b. budgeted cost = -175(5) + 4125 = -875 + 4125 = $3250

c. Each additional day they stay with relatives saves 275 - 100 = $175 per day.

30a. time in minutes = 119n

b. time in hours = 119n _____ 60

c. time = 119n _____ 60

= 119(30)

_______ 60

= 3570 _____ 60

= 59.5 hr

d. time for 1 orbit = 119(1)

______ 60

= 1.98 −

3 h

hours in 1 week = 24 × 7 = 168 h

number of orbits in a week = 168 _____ 1.98

− 3

≈ 84.7 So, almost 85 orbits would be made.

31. x y = -2 x 2 + 5x - 7

-3 -2 (-3) 2 + 5(-3) - 7 = -40

-2 -2 (-2) 2 + 5(-2) - 7 = -25

0 -2 (0) 2 + 5(0) - 7 = -7

2 -2 (2) 2 + 5(2) - 7 = -5

3 -2 (3) 2 + 5(3) - 7 = -10

32. x y = - 3x + 9 ______ x 2 - 1

-3 - 3(-3) + 9

_________ (-3) 2 - 1

= 0

-2 - 3(-2) + 9

_________ (-2) 2 - 1

= -1

0 - 3(0) + 9

_______ (0) 2 - 1

= 9

2 - 3(2) + 9

_______ (2) 2 - 1

= -5

3 - 3(3) + 9

_______ (3) 2 - 1

= - 9 __ 4

33. x y = x 3 - 11x + 1

-3 (-3) 3 - 11(-3) + 1 = 7

-2 (-2) 3 - 11(-2) + 1 = 15

0 (0) 3 - 11(0) + 1 = 1

2 (2) 3 - 11(2) + 1 = -13

3 (3) 3 - 11(3) + 1 = -5

34. A; the minus sign was not distributed in the second step.

35. Distributive Property; possible answer: the Distributive Property says that multiplying by a sum is the same as adding products.

TEST PREP

36. DA) -2x(1 - 3x) = -2x + 6 x 2 B) 2(3x - 1)x = (6x - 2)x = 6 x 2 - 2xC) (3x - 1)2x = 6 x 2 - 2xD) 6x 2 + 2x

37. GF) 12 in. = 1 ftG) 60 min = 1 hH) 7 days = 1 wkJ) 36 in. = 1 yd

38. C3x( y - 1) 2 = 3(4) (3 - 1) 2 = 12 (2) 2 = 12(4)= 48


39. 2a - 5 = 11 ______ + 5 ___ + 5 2a = 16

2a ___ 2 = 16 ___

2

a = 8

40. 2a - 5 = -5 ______ + 5 ___ + 5 2a = 0

2a ___ 2

= 0 __ 2

a = 0

41. 2a - 5 = 39 ______ + 5 ___ + 5 2a = 44

2a ___ 2 = 44 ___

2

a = 22

42. 2a - 5 = 225 ______ + 5 ___ + 5 2a = 230

2a ___ 2

= 230 ____ 2

a = 115



43a. x y =

3(x + 2) 2 ____________

(x - 1)(x - 3)

0 3(0 + 2) 2

____________ (0- 1)(0 - 3)

= 4

1 3(1 + 2) 2

____________ (1 - 1)(1 - 3)

= undefined

2 3(2 + 2) 2

____________ (2 - 1)(2 - 3)

= -48

3 3(3 + 2) 2

____________ (3 - 1)(3 - 3)

= undefined

4 3(4 + 2) 2

____________ (4 - 1)(4 - 3)

= 36

5 3(5 + 2) 2

____________ (5 - 1)(5 - 3)

= 147 ____ 8

b. The expression cannot be evaluated for x = 1, x = 3.

c. { x | x ≠ 1 and x ≠ 3 }

SPIRAL REVIEW 44. triangular prism 45. square pyramid

46. �, �, � 47. �

48. � 49. irrational

50. √ ��

52 ___ 25

= √ ��

4 · 13 _____ 25 · 1

= 2 __ 5 √ �� 13

51. √ �� 24 + √ � 6 = √ �� 4 · 6 + √ � 6 = 2 √ � 6 + √ � 6 = 3 √ � 6

52. 4 √ �� 27 _____ 18

= 4 √ �� 9 · 3

_______ 18

= 4 · 3 √ � 3

_______ 18

= 12 √ � 3

_____ 18

= 2 √ � 3

____ 3

53. √ �� 28 · √ � 7 = √ �� 4 × 7 · √ � 7 = 2 √ � 7 √ � 7 = 2 · 7= 14

1-5 PROPERTIES OF EXPONENTS,PAGES 34–41

CHECK IT OUT!

1a. (2a) 5 = (2a)(2a)(2a)(2a)(2a)

b. 3b 4 = 3 · b · b · b · b

c. -(2x - 1) 3 y 2 = -(2x - 1)(2x -1)(2x - 1) · y · y

2a. ( 1 __ 3 )

-2

= ( 3 __ 1

) 2

= (3) 2 = 9

b. (-5) -5

= 1 _____ (-5) 5

= 1 ___________________ (-5)(-5)(-5)(-5)(-5)

= - 1 _____ 3125

3a. ( 5x 6 ) 3

= 5 3 x (6)(3) = 125 x 18

b. (-2 a 3 b) -3

= 1 ________ (-2 a 3 b) 3

= 1 ___________ (-2) 3 a (3)(3) b 3

= - 1 _____ 8 a 9 b 3

4a. 2.325 × 10 6 __________ 9.3 × 10 9

= 0.25 × 10 -3 = 2.5 × 10 -4

b. (4 × 10 -6 ) (3.1 × 10 -4 ) = 12.4 × 10 -10 = 1.24 × 10 -9

5. speed of light: 3 × 10 5 km ___ s

= 3 × 10 5 km ___ s ( 10 3 m _____ 1 km

) ( 60 s _____ 1 min

)

= 1.8 × 10 10 m ____ min

time = distance _______ speed

= 1.5 × 10 11 m ____________ 1.8 × 10 10 m ___

min

= 0.8 −

3 × 10 min ≈ 8.33 min

THINK AND DISCUSS

1. Possible answer: The Product and Quotient of Powers Properties both require the same base.

2. Possible answer: Move the decimal point so that there is one nonzero digit in front of it. Use the number of places moved for the exponent of 10. If you moved the decimal point left, use a positive exponent. If you moved the decimal point right, use a negative exponent.

3.


1. Possible answer: a number between 1 and 10 multiplied by an integer power of 10.

2. 4 (a - b) 2 = 4(a - b)(a - b)



3. (12xy) 4 = (12xy)(12xy)(12xy)(12xy)

4. - s 3 (-2t) 5 = -s · s · s(-2t)(-2t)(-2t)(-2t)(-2t)

5. (- 1 __ 2 d)

3

= (- 1 __ 2 d) (- 1 __

2 d) (- 1 __

2 d)

6. (- 3 __ 5 )

-2

= (- 5 __ 3 )

2

= (- 5 __ 3 ) (- 5 __

3 )

= 25 ___ 9

7. 5 0 = 1

8. ( 2 __ 3 )

-3

= ( 3 __ 2 )

3

= ( 3 __ 2 ) ( 3 __

2 ) ( 3 __

2 )

= 27 ___ 8

9. 10 -1

= 1 ___ 10

10. (- 3 a 2 b 3 ) 2 = (-3) 2 a (2)(2) b (3)(2) = 9 a 4 b 6

11. c 3 d 2 ( c -2 d 4 )= c d 6

12. 5u v 6 ____ u 2 v 2

= 5 u -1 v 4

= 5v 4 ___ u

13. 10 ( y 5

__ x 2

) 2

= 10 ( y (5)(2)

_____ x (2)(2)

)

= 10y 10

_____ x 4

14. - 2s -3 t(7 s -8 t 5 )= -14 s -11 t 6

= - 14 t 6 ____ s 11

15. -4m (m n 2 ) 3 = -4m( m 3 n (2)(3) )= -4 m 4 n 6

16. (4b) 2

_____ 2b

= 4 2 b 2 ____ 2b

= 16 b 2 ____ 2b

= 8b

17. x -1 y -2

______ x 3 y -5

= x -4 y 3

= y 3

__ x 4

18. (2.2 × 10 5 ) (4.5 × 10 11 ) = 9.9 × 10 16

19. 7.8 × 10 8 __________ 2.6 × 10 -3

= 3 x 10 11

20. 16 × 10 -3 _________ 4.0 × 10 4

= 4 × 10 -7

21. width of hair = 80 microns or 8.0 × 10 -5 m

width of hair _____________________ width of nanoguitar string

= 8.0 × 10 -5 __________ 2.0 × 10 -7

= 4.0 × 10 2 = 4.0 × 100 = 400Therefore, 400 nanoguitar strings would have the same width as a human hair


22. (m + 2n) 3 = (m + 2n)(m + 2n)(m + 2n)

23. 5 x 3 = 5 · x · x · x

24. (-9fg) 3 h 4 = (-9fg)(-9fg)(-9fg) · h · h · h · h

25. 2a (- b 2 - a) 2

= 2a ( -b 2 - a) (- b 2 - a)

26. (-4) -2

= 1 _____ (-4) 2

= 1 ___ 16

27. (- 3 __ 4

) -1

= (- 4 __ 3

) 1

= - 4 __ 3

28. (- 5 __ 2

) -3

= (- 2 __ 5 )

3

= (- 2 __ 5 ) (- 2 __

5 ) (- 2 __

5 )

= - 8 ____ 125

29. - 6 0 = -(1)= -1

30. -100 s 3 t -5 _________ 25 s -2 t 6

= -4 s 5 t -11

= - 4 s 5 ___ t 11

31. (- x 4 y 2 ) 5 = - x (4)(5) y (2)(5) = - x 20 y 10

32. (16 u 4 v 6 ) -2

= 1 _________ (16 u 4 v 6 )

2

= 1 ________ 256 u 8 v 12

33. 8 a 2 b 5 (-2 a 3 b 2 ) = -16 a 5 b 7

34. (3.2 × 10 6 ) (1.7 × 10 -4 ) = 5.44 × 10 2

35. 5.1 × 10 4 __________ 3.4 × 10 -5

= 1.5 × 10 9

36. (6.8 × 10 3 ) (9.5 × 10 5 ) = 64.6 × 10 8 = 6.46 × 10 9

37. 5.02 × 10 11 __________ 5.4 × 10 9

≈ 0.930 × 10 2

= 93 s or 1.55 min



38. 0.00173 g ___

kg = 1.73 × 10

-3

0.02 kg = 2 × 10 -2

Smallest amount of venom that will be fatal to the

mouse is

(1.73 × 10 -3

) (2 × 10 -2

) = 3.46 × 10 -5 g

39. 8 2 = ( 2 3 ) 2 = 2 6 ; 4 1 = 2 2 ;

2 5 ; 16 -2 = ( 2 4 ) -2

= 2 -8 The order is: 16 -2 , 4 1 , 2 5 , 8 2

40. 2 -1 ; -4 3 = - ( 2 2 ) 3 = - 2 6 ;

4 2 = ( 2 2 ) 2 = 2 4 ; 8 -2 = ( 2 3 )

-2 = 2 -6

The order is: - 4 3 , 8 -2 , 2 -1 , 4 2

41. - 8 2 = - ( 2 3 ) 2 = - 2 6 ; 4 0 = ( 2 2 )

0 = 2 0

16 1 = 2 4 = 2 4 ; 2 -2 The order is: - 8 2 , 2 -2 , 4 0 , 16 1

42. 1.3 × 10 15 × 128 = 166.4 × 10 15 = 1.664 × 10 17 oz of water in Lake Michigan The faucet’s leaking rate per year is

1.5 oz ____ min

× 60 min = 90 oz ___ h

90 oz ___ h × 24 h = 2160 oz ____

day

2160 oz ____ day

× 365 days = 788,400 or 7.884 × 10 5 oz ___ yr

The number of years it will take for the amount of water that is leaking to be equal to the amount of water in Lake Michigan is

1.664 × 10 17 ___________ 7.884 × 10 5

≈ 0.211 × 10 12 = 2.11 × 10 11 yr

43. V = � × w × h= m n 2 · m 3 n · 3mn= 3 m 5 n 4

44. V = π r 2 h

= π ( a 2 b) 2 (abc)

= π ( a 4 b 2 ) (abc) = π a 5 b 3 c

45. 27 x 3 y

______ 18 x 2 y 4

= 3x y -3

_____ 2

= 3x ___ 2 y 3

46. ( 3 a 3 b ______ 2 a -1 b 2

) 2

= 3 2 a (3)(2) b 2 ____________ 2 2 a (-1)(2) b (2)(2)

= 9 a 6 b 2 ______ 4 a -2 b 4

= 9 a 8 ___ 4 b 2

47. 12 a 0 b 5 (-2 a 3 b 2 ) = -24 a 3 b 7

48. 72 a 2 b 3 ________ -24 a 2 b 5

= -3 a 0 b -2

= - 3 __ b 2

49. ( 5mn _____ -3 m 2

) -2

= ( -3 m 2 _____ 5mn

) 2

= (-3) 2 m (2)(2)

__________ 5 2 m 2 n 2

= 9 m 4 _______ 25 m 2 n 2

= 9 m 2 ____ 25 n 2

50. 6 x 5 y 3 (-3 x 2 y -1 )= -18 x 7 y 2

51. 1 yd = 36 in. 1 yd 2 = 36 in. × 36 in. 1 yd 2 = 1296 in 2

52. 1 m = 100 cm 1 m 2 = 100 cm × 100 cm 1 m 2 = 10,000 cm 2

53. 1 ft = 12 in. 1 ft 3 = 12 in. × 12 in. × 12 in. 1 ft 3 = 1728 in 3

54. 1 km = 1000 m 1 km 3 = 1000 m × 1000 m × 1000 m 1 km 3 = 10 9 m 3

55a. speed = distance _______ time

= 384,500

_______ 102.75

≈ 3742 km ___ h

b. 3742 km ________ 1 h

= 3742 km ________ 60 min

≈ 62.367 km _________ 1 min

= 62.367 km _________ 60 s

= 1.03945 km ___ s is the speed of Apollo 11.

Future spaceships will travel

3 × 10 5 ________ 1.03945

= 288,608 times as fast as Apollo 11.

c. time = distance _______ speed

= 384,500

_______ 3 × 10 5

≈ 1.28 s

56. -9 a 2 b 6 (-7a b -4 )= 63 a 3 b 2

57. 14 x -2 y 3

________ -8 x -5 y 5

= - 7 x 3 y -2

______ 4

= - 7 x 3 ___ 4 y 2

58. - ( 20 x 6 ____ 2 x 2

) 3

= - 20 3 x (6)(3) ________ 2 3 x (2)(3)

= - 8000 x 18 _______ 8 x 6

= - 1000x 12



59. (10 x -2 y 0 z -3 ) 2

= 100 x -4 z -6

= 100 ____ x 4 z 6

60. (-3 a 2 b -1 ) -3

= 1 __________ (-3 a 2 b -1 )

3

= 1 _______________ (-3) 3 a (2)(3) b (-1)(3)

= 1 _________ -27 a 6 b -3

= - b 3 ____ 27 a 6

61. (8 m 4 n -2 ) (- 3m -2 n) 0

= (8 m 4 n -2 )

= 8 m 4 ____ n 2

62. China;

1.25 × 10 9 _________ 9.60 × 10 6

= 130.2 people

______ mi 2

63. Laos;

6.07 × 10 6 _________ 2.37 × 10 5

= 25.6 people

______ mi 2

64. Thailand;

6.49 × 10 7 _________ 5.14 × 10 5

= 126.3 people

______ mi 2

65. Vietnam;

7.88 × 10 7 _________ 1.28 × 10 5

= 615.6 people

______ mi 2

66. Cambodia;

1.34 × 10 7 _________ 1.81 × 10 5

= 74.0 people

______ mi 2

67. 1.2 beats _____ s × 60 s = 72 beats _____ min

72 beats _____ min

× 60 min = 4320 beats _____ h

4320 beats _____ h × 24 h = 103,680 beats _____

day

103,680 beats _____ day

× 365 day = 37,843,200 beats _____ yr

37,843,200 beats _____ yr × 75 yr = 2,838,240,000 beats ______ lifetime

or ≈ 2.84 × 10 9 beats in 75 years.

68. 16 breaths _______ min

× 60 min = 960 breaths _______ h

960 breaths _______ h × 24 h = 23,040 breaths _______

day

23,040 breaths _______ day

× 365 day = 8,409,600 breaths _______ yr

8,409,600 breaths _______ yr × 75 yr = 630,720,000 breaths _______ lifetime

or ≈ 6.3 × 10 8 breaths in a lifespan of 75 years.

69. 254 hairs _____ cm 2

× 500 cm 2 = 127,000 hairs _____ head

or 1.27 × 10 5 hairs on a human head.

70. Power of a Power Property

71. Power of a Product Property or Power of a Power Property

72. Quotient of Powers Property

73. Power of a Quotient Property or Power of a Power Property

74. 1 million = 10 6 , so 3.8 million = 3.8 × 10 6 .The word million can be represented by the expression 10 6 .

75. Possbile answer: 0 0 = 0 (2 - 2) = 0 2 __ 0 2

= 0 __ 0

but division by zero undefined.

76. (3.7 × 10 -3 ) (8.1 × 10 -5 )

= 2.997 × 10 -7

77. 2.05 × 10 -8 ___________ 3.0 × 10 6

= 6.5 × 10 -15

78. (4.75 × 10 2 ) (4.2 × 10 -7 )

= 1.995 × 10 -4

79. 8.4 × 10 9 __________ 2.4 × 10 -5

= 3.5 × 10 14

80. 17.068 × 10 -4 _____________ 6.8 × 10 3

= 2.51 × 10 -7

81. (1.83 × 10 13 ) (6.2 × 10 10 )

= 1.1346 × 10 24

82. Possible answer: First compare exponents. Since 9 > 8, 1.23 × 10 9 is greater than 4.56 × 10 8 . If the exponents are equal, compare the initial

factors. Since 1.23 < 4.56, 1.23 × 10 7 is less than 4.56 × 10 7 .

TEST PREP

83. C 84. J

85. C 86. J

a 4 b -3 _____ a 2 c 0

= a 2 b -3

= a 2 __ b 3


87. ( 7.82 × 10 6 _________ 5.48 × 10 8

) 2

≈ (1.427 × 10 -2 ) 2

= 1.427 2 × 10 (-2)(2) = 2.0363 × 10 -4

88. ⎡ ⎣ (6.18 × 10 7 ) (2.05 × 10 8 ) ⎤ ⎦ 2

= ( 12.669 × 10 15 ) 2

= (1.2669 × 10 16 ) 2

= 1.2669 2 × 10 (16)(2) ≈ 1.605 × 10 32

89. Possible answer: ( 1 __ 2

) -2

, (0.7) -2 , (- 2 __ 5

) -2

;

numbers between -1 and 1, excluding 0, are greater than 1 when raised to the exponent -2.

90. Possible answer: 2 3 < 3 2 , 1 3 < 3 1 , 0 2 < 2 0 ; 3 4 > 4 3 , 2 5 > 5 2 , 4 5 > 5 4



SPIRAL REVIEW

91. Probability of rock, paper, or scissors is the same:

P (r) = P (p) = P (s) = 1 __ 3 ;

P(same choice) = P (r, r) + P (p, p) + P (s, s)

= 1 __ 3 · 1 __

3 + 1 __

3 · 1 __

3 + 1 __

3 · 1 __

3

= 3 __ 9 = 1 __

3

92. 1 __ 3 · 3 = 1 93. 4(-3 + 8) = -12 + 32

94. 0 = √ � 7 + (- √ � 7 )

95. 2mn ____________ n 2 - 2n + 5m

= 2(3)(-1)

__________________ (-1) 2 - 2(-1) + 5(3)

= -6 __________ 1 + 2 + 15

= -6 ___ 18

= - 1 __ 3

96. 2x (9y - x 2 )

= 2(-3) ⎡ ⎣ 9(10) - (-3) 2 ⎤ ⎦ = -6(90 - 9)= -6(81)= -486

READY TO GO ON? PAGE 43

1. -3 1 __ 3 = -3.3

− 3 ,

√ � 5 ≈ 2.23, - 4 __

5 = -0.8

The order is: -3 1 __ 3 , - 4 __

5 , 0.

−− 75 , √ � 5 , 2.5

-3 1 __ 3 : �, �; - 4 __

5 : �, �;

0. −−

75 : �, �; √ � 5 : �, irrational;

2.5: �,�

2. √ � 3 ≈ 1.732, - π __ 2 ≈ -1.57, 5 __

6 = 0.8

− 3

The order is: -2, - π __ 2 , -1.

−− 15 , 5 __

6 , √ � 3

-2: �, �, �; - π __ 2 : �, irrational;

-1. −−

15 : �, �; 5 __ 6 : �, �;


3. [-4, 2) 4. {x | x < -2 or x > 0}

5. Distributive Property 6. Additive Identity Prop.

7. Assoc. Prop. of Mult.

8. 12% of $250 = (0.12)(250) = (0.1 + 0.02)(250) = (0.1)(250) + (0.02)(250) = 25 + 2(2.5) = 25 + 5 = $30

9. 75 ft 2 dance floor: side size = √ �� 75 ≈ 8.7 ft 125 ft 2 dance floor: side size = √ �� 125 ≈ 11.2 ft 150 ft 2 dance floor: side size = √ �� 150 ≈ 12.2 ft The 75 ft 2 dance floor is the largest that would fit in an 11 ft by 13 ft room.

10. - √ �� 72 = - √ �� 36 · 2 = -6 √ � 2

11. 5 √ �� 12 + 9 √ � 3 = 5 √ �� 4 · 3 + 9 √ � 3 = 5 · 2 √ � 3 + 9 √ � 3 = 10 √ � 3 + 9 √ � 3 = 19 √ � 3

12. -4 √ �� 10 _______

√ � 2

= -4 √ ��

10 ___ 2

= -4 √ � 5

13. √ �� 32 · √ � 6 = √ �� 192 = √ �� 64 · 3 = 8 √ � 3

14. a 2 __ 3 + ab ___

4

= (3) 2

____ 3 +

3(-4) _____

4

= 9 __ 3

- 12 ___ 4

= 3 - 3= 0

15. d 2 ____ 2cd

= (2) 2 _______

2(-1)(2)

= 4 ___ -4

= -1

16. 2 x 2 - 3y + 5x - x 2 = 6 x 2 - 3y

17. 3(x + 2y) - 5x + y= 3x + 6y - 5x + y= -2x + 7y

18. ( x 11 y -2 ) 4

= x (11)(4) y (-2)(4)

= x 44 y -8

= x 44 ____ y 8

19. -3 s 3 t 2 ______ s -2 t 8

= -3 s 5 t -6

= -3 s 5 _____ t 6

20. 4 ( a 2 b 6 ) -3

= 4 _______ ( a 2 b 6 )

3

= 4 _________ a (2)(3) b (6)(3)

= 4 _____ a 6 b 18

21. ( m 4 ________ -5 m -2 n 3

) 2

= m (4)(2) _______________ (-5) 2 m (-2)(2) n (3)(2)

= m 8 ________ 25 m -4 n 6

= m 12 ____ 25 n 6

22. 4.515 × 10 26 ___________ 6.02 × 10 23

= 0.75 × 10 3 = 750 moles

1-6 RELATIONS AND FUNCTIONS,PAGES 44–50

CHECK IT OUT!

1. D: {-2, -1, 0, 1, 2, 3}R: {-3, -2, -1, 0, 1, 2}

2a. function; There is only one price for each size.

b. not a function; Two different carts with the same number of items could cost different amounts.

3a. function

b. not a function; A vertical line can pass through (1, 2) and (1, -2).



THINK AND DISCUSS

1. Possible answer: ordered pairs, table, mapping diagram, and graph

2. Possible answer: any point that is above another on a vertical line has the same input value but a different output value, such as (2, 4) and (2, 1).

3.


1. range 2. D: {0, 1, 2}R: {-2, -1, 0, 1, 2}

3. D: {2000, 2001, 2002, 2003}R: {5.39, 5.65, 5.80, 6.03}

4. function; Each value in the domain is mapped to only one value in the range.

5. not a function; Each value for car model can be mapped to multiple car colors.

6. not a function; Possible answer: a vertical line can be drawn through (2, 1) and (2, 0).

7. function

8. not a function; Possible answer: a vertical line can pass through (2, 2) and (2, -2).


9. D: {Irene, Anna, Lea, Kate}R: {12, 16, 22}

10. D: {-3, 2, 3, 4}R: {-2, 1, 3, 4}

11. function; Each value in the domain is mapped to only one value in the range.

12. not a function; The value 3 is mapped onto two values, 1 and 0.

13. not a function; Possible answer: (1, 1) and (1, -1).

14. function 15. function

16. D: {-5, 0, 5} R: {-5, 0, 5}

17. D: {-2, -1, 0, 1, 2} R: {-2, 0, 2}

18. D: {-2, -1, 1, 3} R: {-3, 0, 3}

19. D: {jumbo, extra large, large, medium}R: {1.75, 2, 2.25, 2.5}

20. D: {e, n, s, v} R: {5, 14, 19, 22}

21a. function; Each year has exactly 5 states for which new quarters are released.

b. function; Each state’s quarter is produced during one year.

c. not a function; Each year releases quarters for 5 different states.

d. function; Each year has exactly 5 states for which new quarters are released.

e. not a function; The number of quarters stays the same each year, while the year changes.

22. D: {-1, 0, 1, 2, 3} R: {-1, 1, 3}

function; For every x-value there is only one y-value.

23. D: {a, b, c, d} R: {1, 2, 4}

function; For each letter there is only one corresponding number

24. D: {7} R: {1, 2, 3, 4, 6} not a function; The domain value 7 is mapped onto

5 range values.

25. D: {1, 3, 5, 7, 9} R: {3} function; For every x-value there is only one y-value.

26. D: {-3, -1, 0, 3} R: {-4, -3, -2, -1, 0} not a function; The value 0 is mapped onto 2 range

values.

27. D: {3, 4, 5, 6, 7}R: {-1, 2, 3}function; For every x-value there is only one y-value.



28. D: {January, February, March, April, May, June, July, August, September, October, November, December}

R: {28, 30, 31}function; Each month in the domain is mapped to one value in the range.

29. D: {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

R: {24}function; Each day has only one number of hours.

30a. a diamond or a square

b. No; (0, 20) and (0, -20) are on the same vertical line.

c. D: {-20, 0, 20} d. R: {-20, 0, 20}

31. B to A is a function; Each person has one date of birth, but a date may have several people born on it.

32. both are functions; Each person has a unique thumbprint and each thumbprint belongs to a unique person.

33. A to B is a function; Each area code belongs to a single state, while a state can have several area codes.

34. both are functions; the amount of sales tax is unique to the purchase total.

35. B to A is a function; The percentage of tax remains the same while the total of the purchase changes.

36. B to A is a function; Each player has a single jersey number, but the same jersey number may belong to different players.

37. Both are functions; Each player has a unique number, and each number belongs to one player.

38. Statement A is incorrect; Possible answer: the input value 0 is paired with 2 output values, which violates the definition of a function.

39. No; the relation is not a function. One input gauge can produce 2 output values.

40. Lengths increase in increments of 1 __ 4 inch;

the relation is a function

41a. Yes, the relation is a function.

b. It is a function; As size increases, number per 1 lb decreases, so average weight increases.

c. size average weight

2d 16 ____ 876

≈ 0.0183 oz

3d 16 ____ 568

≈ 0.0282 oz

4d 16 ____ 316

≈ 0.0506 oz

5d 16 ____ 271

≈ 0.0590 oz

6d 16 ____ 181

≈ 0.0884 oz

42. No; Possible answer: switching the domain and range of the function {(0, 2), (1, 2)} results in {(2, 0), (2, 1)}. There is more than one output for the input 2. The resulting relation is not a function.

43. Possible answer: Set of ordered pairs: Look for a duplicate x-coordinate; Mapping diagram: Look for 2 arrows starting at one domain value; Graph: Use the vertical-line test.

TEST PREP

44. B-2 in the domain is mapped twice.

45. F 46. D


47. for the set to be a function, the elements in the domain cannot be equal, D: {a, -a, 2a, a 2 }, so a ≠ - a ___ + a ___ + a 2a ≠ 0 a ≠ 0

a ≠ 2a ____ - 2a ____ - 2a -a ≠ 0 a ≠ 0

a ≠ a 2 ____ - a 2 ____ - a 2 a(1 - a) ≠ 0 a ≠ 0 or 1 - a ≠ 0 a ≠ 1

-a ≠ 2a ____ - 2a ____ - 2a -3a ≠ 0 a ≠ 0

-a ≠ a 2 ____ - a 2 ____ - a 2 -a(1 + a) ≠ 0 -a ≠ 0 or 1 + a ≠ 0 a ≠ 0 a ≠ 1

2a ≠ a 2 ____ - a 2 ____ - a 2 a(2 - a) ≠ 0 a ≠ 0 or 2 - a ≠ 0 a ≠ 2

a ≠ {-1, 0, 1, 2} and b ∈ �

48. Not one to one; each y-value may correspond to more than one x-value.

49. One to one; each length in feet corresponds to only one length in inches.

50. For the set to be a one-to-one function, the elements

in the range cannot be equal, R: ⎧ ⎨

⎩ b, ab, ab ___

2 ⎫

⎬

⎭ , so

b ≠ ab ___ - b ___ - b 0 ≠ b(a - 1) b ≠ 0 or a - 1 ≠ 0 a ≠ 1

b ≠ ab ___ 2

2b ≠ 2 ( ab ___ 2

)

2b ≠ ab ____ - 2b ____ - 2b 0 ≠ b(a - 2) b ≠ 0 or a - 2 ≠ 0 a ≠ 2

ab ≠ ab ___ 2

2(ab) ≠ 2 ( ab ___ 2

)

2ab ≠ ab ____ - ab ____ - ab ab ≠ 0 a ≠ 0 or b ≠ 0

a ≠ {0, 1, 2} and b ≠ {0}



SPIRAL REVIEW

51. P = 2� + 2w = 2(50) + 2(94) = 100 + 188 = 288 ft

52. A = � × w = 50 × 94 = 4700 ft 2

53. A = π r 2 = π (6) 2 = 36π

≈ 113.1 ft 2

54. √ �� 36 < √ �� 42 < √ �� 49 6 < √ �� 42 < 7 6.4 2 = 40.96 6.5 2 = 42.25 √ �� 42 ≈ 6.5

55. √ �� 16 < √ �� 22 < √ �� 25 4 < √ �� 22 < 5 4.6 2 = 21.16 4.6 2 = 22.09 √ �� 22 ≈ 4.7

56. - √ � 9 < - √ � 8 < - √ � 4 -3 < - √ � 8 < -2 2.8 2 = 7.84 2.9 2 = 8.41 - √ � 8 ≈ -2.8

57. √ �� 81 < √ �� 90 < √ �� 100 9 < √ �� 90 < 10 9.4 2 = 88.36 9.5 2 = 90.25 √ �� 90 ≈ 9.5

58. (-3 y 4 ) 3

= (-3) 3 y (4)(3) = -27 y 12

59. (10 w 2 )

2 _______

5 w 5

= 10 2 w (2)(2) ________ 5 w 5

= 100 w 4 ______ 5 w 5

= 20 ___ w

60. (4 c 6 d 2 ) 2

= 4 2 c (6)(2) d (2)(2)

= 16 c 12 d 4

61. ( x 3 __ z ) 7

= x (3)(7) _____ z 7

= x 21 ___ z 7

1-7 FUNCTION NOTATION, PAGES 51–57

CHECK IT OUT!

1a. f(0) = 0 2 - 4(0) = 0

f ( 1 __ 2 ) = ( 1 __

2 )

2 - 4 ( 1 __

2 )

= - 7 __ 4

f(-2) = (-2) 2 - 4(-2) = 12

b. f(0) = -2(0) + 1 = 1

f ( 1 __ 2 ) = -2 ( 1 __

2 ) + 1

= 0 f(-2) = -2(-2) + 1 = 5

2a. b.

3a. Let x represent the number of pictures, and let f represent the cost of photo processing.

f(x) = 0.27x

b. f(24) = 0.27(24) = $6.48 $6.48 represents cost of processing 24 prints.

THINK AND DISCUSS

1. Possible answer: A reasonable domain is ⎡ ⎢

⎣ 0, 380 ____

156 ⎤ �

⎦ h

because time cannot be negative and the train

completes the trip in 380 ____ 156

h.

2. Possible answer: The name of the function is g, the independent variable is t, and the dependent variable is g(t).

3.


1. independent

2. f(0) = 3(0) - 4 = -4f(1.5) = 3(1.5) - 4 = 0.5 f(-4) = 3(-4) - 4 = -16

3. f(0) = 0 2 + 9

= 9f(1.5) = (1.5) 2 + 9 = 11.25 f(-4) = (-4) 2 + 9 = 25

4. f(0) = 3 (0) 2 - 0 + 2 = 2f(1.5) = 3 (1.5) 2 - 1.5 + 2 = 7.25 f(-4) = 3 (-4) 2 - (-4) + 2 = 54

5. f(0) = 3 f(1.5) = 4 f(-4) = 4

6. f(0) = 1 f(1.5) = 3 f(-4) = 1

7. f(0) = -5 f(1.5) = 1 f(-4) = 1

8. 9.



10.

11. Let x represent the number of living room sets sold. f(x) = 125x

f(50) = 125(50) = 6250This represents the loss, in dollars, if 50 customers purchase the living room set.


12. f(0) = 7(0) - 4 = -4

f ( 3 __ 2 ) = 7 ( 3 __

2 ) - 4

= 13 ___ 2

f(-1) = 7(-1) - 4 = -11

13. f(0) = - (0) 2 + 0 = 0

f ( 3 __ 2 ) = - ( 3 __

2 )

2 + 3 __

2

= - 3 __ 4

f(-1) = - (-1) 2 + (-1) = -2

14. f(0) = -2 (0) 2 + 1 = 1

f ( 3 __ 2 ) = -2 ( 3 __

2 )

2 + 1

= - 7 __ 2

f(-1) = -2 (-1) 2 + 1 = -1

15. f(0) = 2

f ( 3 __ 2 ) = 5

f(-1) = 0

16. f(0) = 4

f ( 3 __ 2 ) = 4

f(-1) = -1

17. f(0) = 0

f ( 3 __ 2 ) = 3

f(-1) = 1 __ 2

18. 19.

20.

21. Let m represent the number of miles over the speed limit.

f(m) = 160 + 4m f(8) = 160 + 4(8) = 192 A fine of $192 for driving 8 mi/h over the speed limit.

22. Let d represnt the depth in feet. P(d) = 14.7 + 0.445d

P(50) = 14.7 + 0.445(50) = 36.9536.95 is the pressure in psi at a depth of 50 ft.

23. f(-3.5) = 3(-3.5) - 6 = -16.5 f(-1) = 3(-1) - 6 = -9

f ( 1 __ 4

) = 3 ( 1 __ 4

) - 6

= -5 1 __ 4

f(2) = 3(2) - 6 = 0 f(11) = 3(11) - 6 = 27

24. f(-8) = -8[1 - 2(-8)] = -136

f ( 2 __ 3

) = ( 2 __ 3 ) ⎡

⎣ 1 - 2 ( 2 __

3 ) ⎤

⎦

= - 2 __ 9

f(1) = 1[1 - 2(1)] = -1 f(9) = 9[1 - 2(9)] = -153 f(4) = 4[1 - 2(4)] = -28

25. f(-4) = 2(-4) - 1

_________ 3

= -3

f(0) = 2(0) - 1

_______ 3

= - 1 __ 3

f ( 1 __ 2

) = 2 ( 1 __

2 ) - 1 ________

3

= 0

f(5) = 2(5) - 1

_______ 3

= 3

26. f(-6) = (-6 - 1) 2 + 4 = 53

f (- 3 __ 2

) = (- 3 __ 2

- 1) 2 + 4

= 10 1 __ 4

f(1) = (1 - 1) 2 + 4 = 4 f(4) = (4 - 1) 2 + 4 = 13

27. f(-2) = -1 f(-1) = 2 f(1) = 2 f(2) = -1

28. f (- 3 __ 2

) = 2

f(-1) = 1 f(0) = -1

f ( 1 __ 2

) = -2

29. D: {A | A ≥ 0}R: {y | y ∈ �}Possible answer: For every area, there is only 1 appropriate number of boxes of tile.

30. D: {h | h ∈ �}R: {y | y ≥ 0 and y is a multiple of 4}Possible answer: The situation represents a function because each horse needs exactly 4 shoes.

31. D: {t | t ≥ 0}R: {y | -16 < y ≤ 32.8}Possible answer: For every time, the diver can be in only one place.

32. Possible answer: D: {h | h ≥ 0} R: {y | -130 < y < 60}For every time, there is only one temperature reading for a given thermometer.

33. t = 35; the number of years it takes for plan h to reach a value of $7500.

34. h(25) ≈ 4400; g(25) ≈ 3500

35. t = 40; the time when plan g is worth half the value of plan h.



36. h: 12 years; g: 10 years

37. h(40) - g(40) = 5000; the difference in the plan after 40 years.

38a. c(125) = 175 + 3.5(125) = 175 + 437.5 = 612.5 or $612.50

b. 450 = 175 + 3.5p _____ - 175 ___________ - 175 275 = 3.5p

275 ____ 3.5

= 3.5p

____ 3.5

78.6 ≈ p 78 pots can be produced for $450.

c. a line starting at (0, 175) and rising to the right

39. When x = 3, f(x) = 1 _____ x - 3

= 1 __ 0 , but division by 0 is

undefined.

40. For -5 < x < 1, g(x) is the square root of a negative number, which is not defined for real numbers.

41. For -5 < x < 0, x represents negative hours, and distance traveled would be negative.

42. (2, 8) and (3, 11)

43. independent: number of shirts;dependent: total cost;D: { x | x ≥ 15}

44. independent: hospital charges;dependent: the amount Belinda pays;D: { x | x ≥ 0}

45. f(x) = 2.37x

46. f(x) = 7.5x

47. f(x) = 0.8x

48. f(x) = 250 + 0.05x

49. Possible answer: A domain and range are reasonable if they make sense for the problem. For example, a domain that includes negative values is not reasonable for a problem involving the number of boxes of kitchen tile required to cover a floor with area A.

TEST PREP

50. C f(1) = 2; g(1) = 14; so f(1) < g(1)

51. H h(3) = 15; h(1) = 15; so h(3) = h(1)

52. D f(1) = -3 (1) 2 + 12 = 9 f(3) = -3 (3) 2 + 12 = -15 f(4) = -3 (4) 2 + 12 = -36 f(9) = -3 (9) 2 + 12 = -231 f(10) = -3 (10) 2 + 12 = -288 f(x) = 9 when x = 1.

53. f(-1) = 3 (-1 - 2) 2 + 4 = 3 (-3) 2 + 4 = 3(9) + 4 = 27 + 4 = 31


54. f(2c) = √ �� (2c) 3

= √ �� (2c)(2c)(2c)

= √ �� 4 c 2 · 2c = 2c √ � 2c

55. g (- h __ 4

) = 6 (- h __

4 ) + h _________

2 (- h __ 4

)

= - 6h ___

4 + h ________

- 2h ___ 4

= - 2h ___ 4

÷ - 2h ___ 4

= - 2h ___ 4

× - 4 ___ 2h

= 1



56. h ( t 2 + 3t) = 4 ( t 2 + 3t) + 7t = 4 t 2 + 12t + 7t = 4 t 2 + 19t

57. r ( t 4 ) = √ ��

( t 4 ) 2 + ( 2 __

t 4 )

2

= √ ��

t 8 + 4 __ t 8

= √ ��

t 16 ___ t 8

+ 4 __ t 8

= √ ��

t 16 + 4 ______

t 8

=

√ �� t 16 + 4

________

√ � t 8

=

√ ��

t 16

+ 4 ________

t 4

58a. Yes; for each value of h, there is only one value of A.

b. function; possible answer: any combination of values for b and h gives only one possible value

of A.

SPIRAL REVIEW

59. 4(x + 2) - x(y - 8)= 4x + 8 - xy + 8x= 12x - xy + 8

60. (2a) 2 + 6 a 2 = 4 a 2 + 6 a 2 = 10 a 2

61. 3c - 10 + 2c ___________ 5c

= 5c - 10 _______ 5c

= 5(c - 2)

_______ 5c

= c - 2 _____ c

62. s(s + 7) - 4s= s 2 + 7s - 4s= s 2 + 3s

63. b is any value. 64. b ≠ -3, 0, or 5

65. function

66. not a function; All x-value inputs are the same.

1-8 EXPLORING TRANSFORMATIONS,PAGES 59–66

CHECK IT OUT!

1a. (3, 3) b. (-2, 1)

2a. x + 3 x y

-2 + 3 = 1 -2 4

-1 + 3 = 2 -1 0

0 + 3 = 3 0 2

2 + 3 = 5 2 2

b. x y -y

-2 4 -1(4) = -4

-1 0 -1(0) = 0

0 2 -1(2) = -2

2 2 -1(2) = -2

3. x y 2y

-1 3 2(3) = 6

0 0 2(0) = 0

2 2 2(2) = 4

4 2 2(2) = 4

4. The transformation will be a vertical compression by

a factor of 3 __ 4 .

THINK AND DISCUSS

1. Possible answer: translation 2 units left or horizontal

compression by a factor of 1 __ 2

2. Possible answer: both squeeze the graph toward the y-axis. In a vertical stretch, the y-coordinates change. In a horizontal compression, thex-coordinates change.

3.


1. compression 2. (-1, 2)

3. (4, -1) 4. (5, 8)



5. x y y + 2

-2 1 1 + 2 = 3

0 1 1 + 2 = 3

1.5 0 0 + 2 = 2

3 -2 -2 + 2 = 0

5 0 0 + 2 = 2

6. -x x y

-1(-2) = 2 -2 1

-1(0) = 0 0 1

-1(1.5) = -1.5 1.5 0

-1(3) = -3 3 -2

-1(5) = -5 5 0

7. x y -y

-2 1 -1(1) = -1

0 1 -1(1) = -1

1.5 0 -1(0) = 0

3 -2 -1(-2) = 2

5 0 -1(0) = 0

8. 3x x y

3(-4) = -12 -4 1

3(-3) = -9 -3 0

3(-1) = -3 -1 2

3(0) = 0 0 1

3(1) = 3 1 2

3(3) = 9 3 0

3(4) = 12 4 1

9. x y 3y

-4 1 3(1) = 3

-3 0 3(0) = 0

-1 2 3(2) = 6

0 1 3(1) = 3

1 2 3(2) = 6

3 0 3(0) = 0

4 1 3(1) = 3

10. x y 1 __ 3

y

-4 1 1 __ 3

(1) = 1 __ 3

-3 0 1 __ 3 (0) = 0

-1 2 1 __ 3

(2) = 2 __ 3

0 1 1 __ 3

(1) = 1 __ 3

1 2 1 __ 3

(2) = 2 __ 3

3 0 1 __ 3 (0) = 0

4 1 1 __ 3

(1) = 1 __ 3

11. vertical compression by a factor of 1 __ 2



12. vertical shift up 1.5 units

13. horizontal shift right 5 units


14. (5, 1) 15. (3, 5)

16. (-2, -3)

17. x y y - 2

-3 2 2 - 2 = 0

-1 0 0 - 2 = -2

0 1 1 - 2 = -1

1 0 0 - 2 = -2

3 2 2 - 2 = 0

18. x y -y

-3 2 -1(2) = -2

-1 0 -1(0) = 0

0 1 -1(1) = -1

1 0 -1(0) = 0

3 2 -1(2) = -2

19. x + 3 x y

-3 + 3 = 0 -3 2

-1 + 3 = 2 -1 0

0 + 3 = 3 0 1

1 + 3 = 4 1 0

3 + 3 = 6 3 2

20. -x x y

-1(-3) = 3 -3 2

-1(-1) = 1 -1 0

-1(0) = 0 0 1

-1(1) = -1 1 0

-1(3) = -3 3 2

21. x y 2 __ 3

y

-3 2 2 __ 3

(2) = 4 __ 3

-1 0 2 __ 3 (0) = 0

0 1 2 __ 3

(1) = 2 __ 3

1 0 2 __ 3 (0) = 0

3 2 2 __ 3 (2) = 4 __

3



22. 1 __ 2 x x y

1 __ 2 (-3) = - 3 __

2 -3 2

1 __ 2 (-1) = - 1 __

2 -1 0

1 __ 2 (0) = 0 0 1

1 __ 2 (1) = 1 __

2 1 0

1 __ 2 (3) = 3 __

2 3 2

23. 3 __ 2 x x y

3 __ 2 (-3) = - 9 __

2 -3 2

3 __ 2 (-1) = - 3 __

2 -1 0

3 __ 2 (0) = 0 0 1

3 __ 2 (1) = 3 __

2 1 0

3 __ 2 (3) = 9 __

2 3 2

24. x y 2y

-3 2 2(2) = 4

-1 0 2(0) = 0

0 1 2(1) = 2

1 0 2(0) = 0

3 2 2(2) = 4

25. vertical shift down 5 units


27. horizontal stretch by a factor of 2

28. 10 square units; the same as the original


30. 20 square units; larger than the original

31. 7 square units; smaller than the original

32. 7 square units; smaller than the original



35. 30 square units; larger than the original

36a. a horizontal shift 10 units right or a vertical shift 30 units down

b. possible answers: f(x) = 3(x - 10)

37a. vertical translation

b. horizontal compression

c. the increase in the per-hour labor rate 60 + 65(3) = $255 50 + 75(3) = $275



38a. They are both linear graphs.

b. The graphs are parallel lines.

c. Add 10 to f or subtract 10 from g.

d. The graph is 10 units above until x = 150,then 10 units below.

39. 40.

41.

42. Roberta started half an hour later.

43. The library is half as far from Roberta’s house.

44. Possible answer: Order is important in these transformations: horizontal translation and reflection across the y-axis; vertical translation and reflection across the x-axis. Order is not important in these transformations: horizontal translation and reflection across the x-axis; vertical translation and reflection across the y-axis.

45. Possible answer: You might not need to make a table of values to graph a transformation of a function. For example, if the graph of a function is translated 2 units right, you can graph the transformation by shifting each point on the graph of the original function 2 units right.

TEST PREP

46. D(x, y) → (x, ay).

47. H (x, y) → (-x, y)

48. D(x, y) → (bx, y)

49. H (x, y) → (-x, y)

50. D

51. Possible answer: Translate down 6 units, or reflect across the x-axis.


52. 2x = 22

x __ 2 = 22 ___

2

x = 11

y - 3 = 7 _____ + 3 ___ + 3 y = 10

The original point was (11, 10).

53a. c(n) = 0.37n b. Vertical stretch

c. 15 in 1999 and 13 in 2002.

d. The number of letters that can be mailed for $5.00 must be rounded down to the nearest whole number.

54. for (x, -y) = (-x, y) x = -x 2x = 0 x = 0

y = -y 2y = 0 y = 0

(0, 0) is the only point that satisfies this condition.

SPIRAL REVIEW

55. 172 + 150 + x ____________ 3

= 144

322 + x _______ 3

= 144

3 ( 322 + x _______ 3

) = 3(144)

322 + x = 432 _________ - 322 _____ - 322 x = 110

56. function 57. function



58. not a function 59. f(1) = 4(1) - 5

_______ 2

= - 1 __ 2

f(-3) = 4(-3) - 5

_________ 2

= - 17 ___ 2

f ( 1 __ 4 ) =

4 ( 1 __ 4

) - 5 ________

2

= -2

60. f(1) = 2 (1) 3 = 2 f(-3) = 2 (-3) 3 = -54

f ( 1 __ 4 ) = 2 ( 1 __

4 )

3

= 1 ___ 32

61. f(1) = [1 - (1) 2 ] 2 = 0

f(-3) = [1 - (-3) 2 ] 2 = 64

f ( 1 __ 4 ) = ⎡

⎣ 1 - ( 1 __

4 )

2 ⎤ ⎦ 2

= 225 ____ 256

1-9 INTRODUCTION TO PARENT FUNCTIONS, PAGES 67–73

CHECK IT OUT!

1a. g(x) = x 3 + 2 is cubic. g(x) = x 3 + 2 represents a vertical translation of the

cubic parent function 2 units up.

b. g(x) = (-x) 2 is quadratic. g(x) = (-x) 2 represents a reflection of the quadratic

parent function across the y-axis.

2. The data points resemble a linear function. The data set is a vertical stretch of the linear parent

function by a factor of 3.

126

12

6

-12 -6

3. The graph resembles a linear function. The cost for 5 months of online services is about

$72.

0 3 6 9 12

30

60

90

120

Time (mo)

Cost

($)

THINK AND DISCUSS

1. Possible answer: Look at its function rule or sketch the graph of the function to see the shape.

2. Possible answer: Recognizing the parent function can help you predict what the graph will look like and help you fill in the missing parts.

3.


1. Possible answer: Within a family of functions, each function is a transformation of the parent function.

2. g(x) = (x - 1) 3 is cubic. g(x) = (x - 1) 3 represents a translation of the cubic

parent function 1 unit right.

3. g(x) = (x + 1) 2 is quadratic. g(x) = (x + 1) 2 represents a translation of the

quadratic parent function 1 unit left.

4. g(x) = -x is linear. g(x) = -x represents a reflection of the linear parent

function across the y-axis.

5. g(x) = √ �� x + 3 is a square root. g(x) = √ �� x + 3 represents a translation of the

square root parent function 3 units left.

6. g(x) = x 2 + 4 is quadratic. g(x) = x 2 + 4 represents a translation of the

quadratic parent function 4 units up.

7. g(x) = x - √ � 2 is linear.

g(x) = x - √ � 2 represents a translation of the linear

parent function √ � 2 units down.

8. The data points resemble a linear function. The data set is a vertical stretch or horizontal

compression of the linear parent function by a factor of 5 or 1 __

5 , respectively.



9. The data points resemble a cubic function. The data set is a vertical compression or horizontal

stretch of the cubic parent function by a factor of 1 ___ 27

or 27, respectively.

10a.

b. The graph resembles the shape of a square root parent function.

c. The string length must be about 5 m to have a complete swing of 4.5 s.

d. It takes about 7.5 s to complete a swing if the string length is 14 m.


11. g(x) = x 2 - 1 is quadratic. g(x) = x 2 - 1 represents a translation of the

quadratic parent function 1 unit down.

12. g(x) = √ �� x - 2 is a square root. g(x) = √ �� x - 2 represents a translation of the

square root parent function 2 units right.

13. g(x) = x 3 + 3 is cubic. g(x) = x 3 + 3 represents a translation of the cubic

parent function 3 units up.

14. The data points resemble a quadratic function. The data set is a vertical compression or horizontal

stretch of the quadratic parent function by a factor

of 1 __ 3 or 3, respectively.

15. The data points resemble a square root function. The data set is a vertical stretch or horizontal

compression of the quadratic parent function by a factor of 2 or 1 __

2 , respectively.

16a.

b. quadratic parent function

c. 10 points d. 21 segments

17. D: {x | x ≥ 0}; R: {y | y ≥ 0}; vertical stretch by a factor of 3

18. D: {x | x ∈ �}; R: {y | y ∈ �}; vertical compression by a factor of 2 __

3

19. D: {x | x ≥ 0}; R: {y | y ≤ 0}; reflection across the x-axis

20. D: {x | x ∈ �}; R: {y | y ≤ 0}; horizontal shift right 2 units and then reflection

across the x-axis

21. D: {x | x ∈ �}; R: {y | y ≤ 1}; reflection across the x-axis and then a vertical shift

up 1 unit



22. D: {x | x ∈ �}; R: {y | y ∈ �}; reflection across the x-axis and a vertical

compression by a factor of 1 __ 2

23. The total cost of 15 tickets is $195; possible answer: cost could be determined by estimating from a graph of the data in the table.

24. The data set is a reflection of the cubic parent function across the x-axis or reflection across the y-axis.

25. The data set is a translation of the quadratic parent function by 7 units right.

26. The data set is a reflection of the square root parent function across the y-axis.

27. The data set is a reflection of the linear parent function across y-axis and a vertical shift down by 1 unit.

28a. linear function b. quadratic function

c. square root function

29. linear function; The width of a photo 1000 pixels high is about 1500 pixels.

30. linear function; The height of a photo 500 pixels wide is about 334 pixels.

31. quadratic function; The width of a photo with a file size of 1000 KB is about 1417 pixels.

32. linear function; D: {h | h ≥ 0}; R: {y | y ≥ 0}; Unlike the linear parent function, the domain and

range of this situation do not include the negative values in �.



33. cubic function; D: {� | � ≥ 0}; R: {y | y ≥ 0}; Unlike the cubic parent function, the domain this

situation does not include the negative values in �.

34. linear function; D: {w | w ≥ 0}; R: {y | y ≥ 0}; Unlike the linear parent function, the domain and

range of this situation do not include the negative values in �.

35. linear function; D: {n | n ∈ �}; R: {y | y ∈ �}; Unlike the linear parent function, the domain and

range include only values in �.

36. linear function; D: {p | p ≥ 0}; R: {y | y ≥ 0}; Unlike the linear parent function, the domain and

range in this situation do not include the negative values in �.

37. square root function; D: {a | a ≥ 0}; R: {y | y ≥ 0}; Same domain and range as parent function.

38. The volume of 1 g of aerogel is about 333 cm 3 .

39a. linear function b. cubic function

c. quadratic function d. square root function

e. linear function; horizontal stretch by a factor of 2 and a vertical shift up 3 units

40. Possible answer: A horizontal translation results from a constant being added to x before squaring,

such as (x + a) 2 . A vertical translation results from a constant being added to x 2 , such as x 2 + a. A reflection across the x-axis results from negating x 2 , such as -x 2 .

41. Constant, square root, linear, quadratic, cubic; the constant function does not increase at all; the

square root function increases slowly; the linear function increases 1 to 1 as x increases; the quadratic and cubic functions increase quickly, with cubic being the faster of the 2.

TEST PREP

42. D 43. H

44. B x ≠ 0; x 2 > 0; - x 2 < 0

45. G

46. D


47. quadratic function, since highest power of x is 2

48. constant function, since h(x) = 1 + 2 = 3 and the highest power of x is 0

49. linear function, since highest power of x is 1

50a. b. D: {x | x ∈ �}; R: {y | y > 0}



c. f(0) = 2 0 = 1; So, function crosses y-axis at (0, 1).

d. (0, 1); Possible answer: 3 0 = 1, so f(x) = 3 x will have the same y-intercept as f(x) = 2 x .

SPIRAL REVIEW

51. (1.5 × 10 -4 ) (5.0 × 10 13 ) = 7.5 × 10 9

52. (8.1 × 10 3 ) 2

= 65.61 × 10 6 = 6.561 × 10 7

53. 1.9 × 10 -6 __________ 9.5 × 10 18

= 0.2 × 10 -24 = 2.0 × 10 -25

54. f(-3) = 1 __ 2 (-3) + 3

= 3 __ 2

f(0) = 1 __ 2 (0) + 3

= 3

f ( 1 __ 3 ) = 1 __

2 ( 1 __

3 ) + 3

= 19 ___ 6

f(6) = 1 __ 2 (6) + 3

= 6

55. f(-5) = (-5)(-5 + 2) = 15

f (- 2 __ 3 ) = (- 2 __

3 ) (- 2 __

3 + 2)

= - 8 __ 9

f(1.6) = (1.6)(1.6 + 2) = 5.76 f(4) = (4)(4 + 2) = 24

56. (1, 1) 57. (4, -10)

58. (-3, -5)

READY TO GO ON? PAGE 75

1. D: {x | -3 ≤ x ≤ 3} R: {y | -1 ≤ y ≤ 2} not a function

2. D: {0, 2, 4, 6} R: {5, 8, 10, 12, 20} not a function

3. D: {x | -1 ≤ x ≤ 3} R: {y | 0 ≤ y ≤ 4} function

4. f(0) = 12 - 3(0) = 12 f(1) = 12 - 3(1) = 9 f(-2) = 12 - 3(-2) = 18

5. f(0) = 3 (0) 3 + 1 = 1 f(1) = 3 (1) 3 + 1 = 4 f(-2) = 3 (-2) 3 + 1 = -23

6. f(0) = 4 - (0) 2 = 4 f(1) = 4 - (1) 2 = 3 f(-2) = 4 - (-2) 2 = 0

7a. Let m represent the distance driven in miles and let c represent the cost per mile in dollars, c(m) = 1.75 + 0.25(4m) = 1.75 + m

b.

c. c(5.5) = 1.75 + 5.5 = 7.25 It represents the cost in dollars for a taxi ride of

5.5 miles.

8. vertical translation up by 15 units

9. vertical compression by a factor of 0.6

10. g(x) = - x 2 is quadratic. g(x) = - x 2 represents a reflection of the quadratic

parent function across the x-axis.

11. g(x) = √ �� x - 3 is a square root.

g(x) = √ �� x - 3 represents a translation of the

square root parent function 3 units right.

12. g(x) = 1.5x is linear. g(x) = 1.5x represents a vertical stretch of the linear

parent function by a factor of 1.5.



13. quadratic; A stand with a maximum load of 7920 kg has a diameter of about 17.5 mm.

STUDY GUIDE: REVIEW, PAGES 76–79

1. domain; range

LESSON 1-1

2. {x | x ≥ -5} 3. (1, 5]

4. {4, 5, 6, 7, …} 5. {x | x < -2 or x > 5}

6. integers greater than -4 and less than or equal to 5

7. [5.5, 5.6]

LESSON 1-2

8. Comm. Prop. of Mult. 9. Distributive Property

10. -0.55; 1 ____ 0.55

11. 7 __ 8 ; - 8 __

7

12. -1. −

2 ; 1 ___ 1.

− 2 or 9 ___

11

LESSON 1-3

13. √ � 9 <

√ �� 12 <

√ �� 16

3 < √ �� 12 < 4

3.4 2 = 11.56 3.5 2 = 12.25

√ �� 12 ≈ 3.5

14. √ �� 49 <

√ �� 55 <

√ �� 64

7 < √ �� 55 < 8

7.4 2 = 54.76 7.5 2 = 56.25

√ �� 55 ≈ 7.4

15. √ �� 64 <

√ �� 74 <

√ �� 81

8 < √ �� 74 < 9

8.6 2 = 73.96 8.7 2 = 75.69

√ �� 74 ≈ 8.6

16. √ �� 25 <

√ �� 29 <

√ �� 36

5 < √ �� 29 < 6

5.3 2 = 28.09 5.4 2 = 29.16

√ �� 29 ≈ 5.4

17. √ �� 32 = √ �� 16 · 2 = 4 √ � 2

18. √ �� 64

____ √ � 4

= 8 __ 2

= 4

19. 2 √ � 2 - √ �� 72 = 2 √ � 2 - √ �� 36 · 2 = 2 √ � 2 - 6 √ � 2 = -4 √ � 2

20. √ � 3 · √ �� 21 = √ �� 63 = √ �� 9 · 7 = 3 √ � 7

21. 7 ___ √ � 2

= 7 ___ √ � 2

· √ � 2 ___ √ � 2

= 7 √ � 2 ____ √ � 4

= 7 √ � 2 ____ 2

22. 2 √ �� 20 _____

5 √ � 8

= 2 √ �� 4 · 5

_______ 5 √ �� 4 · 2

= 2 · 2 √ � 5

_______ 5 · 2 √ � 2

= 2 √ � 5

____ 5 √ � 2

= 2 √ � 5

____ 5 √ � 2

· √ � 2 ___ √ � 2

= 2 √ �� 10

_____ 5 √ � 4

= 2 √ �� 10

_____ 5 · 2

= √ �� 10

____ 5

LESSON 1-4

23. x 2 y - x y 2 = (6) 2 (-2) - (6) (-2) 2 = 36(-2) - 6(4)= -72 - 24= -96

24. - x 2 __ 2 + 5xy - 9y

= - (4) 2

____ 2

+ 5(4)(2) - 9(2)

= - 16 ___ 2 + 40 - 18

= -8 + 40 - 18 = 14

25. n 2 + mn - 1 ___________ 4 m 2 n

= (-1) 2 + (2)(-1) - 1

_________________ 4 (2) 2 (-1)

= 1 - 2 - 1 _________ 4(4)(-1)

= -2 ____ -16

= 1 __ 8

26. -x - 2y + 9x - y + 3x= -x + 9x + 3x - 2y - y= 11x - 3y

27. 7 - (5a - b) + 11= 7 - 5a + b + 11= 7 + 11 - 5a + b= 18 - 5a + b

28. -4(2x + 3y) + 5x= -8x - 12y + 5x= -8x + 5x - 12y= -3x - 12y

29. c ( a 2 - b) + 3bc= a 2 c - bc + 3bc= a 2 c + 2bc



LESSON 1-5

30. (-2 x 5 y -3 ) 3 = -8 x 15 y -9

= -8 x 15 ______ y 9

31. -24 x 4 y -6

_________ 14x -3 y 3

= -12 x 7 y -9

_________ 7

= -12 x 7 ______ 7 y 9

32. ( r 2 s ___ s 3

) 2

= r 4 s 2 ____ s 6

= r 4 __

s 4

33. 4mn ( m 5 n -5 ) = 4 m 6 n -4

= 4 m 6 ____ n 4

34. 7.7 × 10 5 __________ 1.1 × 10 -2

= 7 × 10 7

35. (4.5 × 10 -2 ) (1.2 × 10 3 ) = 5.4 × 10 1

LESSON 1-6

36. D: {3, 5, 7} R: {-1, 0, 9} not a function

37. D: [-2, ∞) R: [-4, ∞) not a function

38. D: {-2, 0, 3, 4} R: {3, 4} function

39. D: {5, 10, 15, 20, 25} R: {-5, -4, -3, -2, -1} function

40. D: {a, b, c} R: {Alabama, Alaska, Arizona, Arkansas, California,

Colorado, Connecticut} not a function

LESSON 1-7

41. f(2) = - (2) 2 + 2 = -2

f ( 1 __ 2 ) = - ( 1 __

2 )

2 + 2

= 7 __ 4

f(-2) = - (-2) 2 + 2 = -2

42. f(2) = -5(2) - 6 = -16

f ( 1 __ 2 ) = -5 ( 1 __

2 ) - 6

= - 17 ___ 2

f(-2) = -5(-2) - 6 = 4

43. f(2) = -1

f ( 1 __ 2 ) = 1

f(-2) = 2

44. f(2) = 1 __ 2

f ( 1 __ 2 ) = 2

f(-2) = - 1 __ 2

45. 46.

47. A(s) = 6 s 2 , where A is the surface area in square units and s is the side length in linear units.

A(10) = 600; the surface area of a cube of side length 10 cm is 600 cm 2 .

LESSON 1-8

48. (0, -5) 49. (5, 1)


Parking Fees

0 1 2 3

4

8

12

Time (h)

Fee

($)

4 5

51. vertical stretch by a factor of 1.1

Parking Fees

0 1 2 3

4

8

12

Time (h)

Fee

($)

4 5

52. translation 1 unit up Parking Fees

0 1 2 3

4

8

12

Time (h)

Fee

($)

4 5

LESSON 1-9

53. quadratic function; translation 1 unit down

54. square root function; reflection across the x-axis

55. linear function; For a 95 lb rider, the tire pressure is about 90 psi.



CHAPTER TEST, PAGE 80

1. - √ � 3 ≈ -1.73

The order is: -2, - √ � 3 , 0.95, 1, 1. −

5 -2: �, �, �; - √ � 3 : �, irrational; 0.95: �, �; 1: �, �, �, , ; 1.

− 5 : �, �

2. (-∞, -2) and (1, 3] 3. {x | x ≤ 12}

4. Comm. Prop. of Add. 5. Distributive Property

6. Multiplicative Identity Property

7. √ � 6 ≈ 2.4 ft; √ � 8 ≈ 2.8 ft; √ �� 15 ≈ 3.9 ft;The 8 ft 2 window is the largest that could fit in the wall.

8. -2 √ � 3 + √ �� 75 = -2 √ � 3 + √ �� 25 · 3 = -2 √ � 3 + 5 √ � 3 = 3 √ � 3

9. √ �� 24 - √ �� 54 = √ �� 4 · 6 - √ �� 9 · 6 = 2 √ � 6 - 3 √ � 6 = - √ � 6

10. √ �� 22 · √ �� 55 = √ �� 1210 = √ �� 121 · 10 = 11 √ �� 10

11. 2(x + 1) + 9x= 2x + 2 + 9x= 11x + 2

12. 5x - 5y - 7x + y= -2x - 4y

13. 12x + 4(x + y) - 6y= 12x + 4x + 4y - 6y= 16x - 2y

14. 8 a 2 b 5 (-2 a 3 b 2 ) = -16 a 5 b 7

15. 28 u -2 v 3 _______ 4 u 2 v 2

= 7 u -4 v

= 7v ___ u 4

16. (5 x 4 y -3 ) -2

= 5 -2 x (4)(-2) y (-3)(-2)

= 5 -2 x -8 y 6

= y 6 ____

25 x 8

17. ( 3 x 2 y ____

x y 2 )

-1

= x y 2

____ 3 x 2 y

= y ___

3x

18. 2.25 × 10 8 _________ 5 × 10 6

= 0.45 × 10 2 = 45

19. D: {8, 9, 10}R: {2, 4, 6, 8, 10}not a function

20. D: [-5, 5]R: [-2, 2]function

21. f(-2) = -4(-2) = 8

f ( 1 __ 2 ) = -4 ( 1 __

2 )

= -2 f(0) = -4(0) = 0

22. f(-2) = -3 (-2) 2 + (-2) = -14

f ( 1 __ 2

) = -3 ( 1 __ 2 )

2 + 1 __

2

= - 1 __ 4

f(0) = -3 (0) 2 + (0) = 0

23. f(-2) = √ �� (-2) + 3 = 1

f ( 1 __ 2

) = √ ��

( 1 __ 2

) + 3

= √ �� 3.5 ≈ 1.87 f(0) = √ �� 0 + 3 = √ � 3 ≈ 1.73

24. square root function; For a building with height of 80 m, the distance to

the horizon is about 32 km.

0 20 40 60 80

8

16

24

32

Height of building (m)

Dis

tanc

e to

hor

izon

(km

)

100



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Ch 1 Solution Key