prprecalculus with limits, answers to section 1.1 1 · precalculus with limits, answers to section...

PrPrecalculus with Limits, Answers to Section 1.1 1

Chapter 1Section 1.1 (page 9)

Vocabulary Check (page 9)1. (a) v (b) vi (c) i (d) iv (e) iii (f) ii2. Cartesian 3. Distance Formula4. Midpoint Formula

1. A: B: C: D: 2. A: B: C: D:3. 4.

5. 6.

7. 8. 9.

10. 11. Quadrant IV 12. Quadrant III

13. Quadrant II 14. Quadrant I

15. Quadrant III or IV 16. Quadrant I or IV

17. Quadrant III 18. Quadrant III

19. Quadrant I or III 20. Quadrant II or IV

21. 22.

23. 8 24. 7 25. 5 26. 10

27. (a) 4, 3, 5 (b)

28. (a) 5, 12, 13 (b)

29. (a) (b)

30. (a) (b)

31. (a) (b) 10(c)

32. (a) (b) 13(c)

33. (a) (b) 17(c)

34. (a) (b) 15(c) ��5

2, 2�

x

(2, 8)8

6

2

−2−2−6−8−10

−4

2

y

(−7, −4)

�0, 52�

x

(−4, 10)

(4, −5)

4−8 −6 −4 −2 6 8

−6

−4

2

6

8

10

y

�72, 6�

x(6, 0)

(1, 12)12

10

8

6

4

2

2−2 4 6 8 10

y

�5, 4�

x(1, 1)

(9, 7)

12

10

8

6

4

2

2−2 4 6 8 10

y

42 � 72 � ��65 �24, 7, �65

102 � 32 � ��109 �210, 3, �109

52 � 122 � 132

42 � 32 � 52

Tem

pera

ture

(in

°F)

Month (1 ↔ January)

x

−40

10

−30

−20

−10

20

30

2 6 8 10 12

40

0

yy

x6 7 8 9 10 11 12 13

3000

3500

4000

4500

5000

Year (6 ↔ 1996)

Num

ber

of s

tore

s

��12, 0��5, �5��4, �8��3, 4�

−1−2−3 2 3−1

−2

2

1

3

4

y

x

y

x−2−4−6 2 4 6 8

−4

−6

2

4

6

8

−1−2−3 1 2 3−1

−2

2

1

3

4

y

x−2−4−6 2 4 6

−2

−4

−6

2

4

6

y

x

��6, 0��3, 52�,�0, �2�,� 32, �4�,

��3, 2��4, �4�,��6, �2�,�2, 6�,

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333202CB01a_AN.qxd 4/26/06 6:30 PM

(Continued)

35. (a) (b)(c)

36. (a) (b)(c)

37. (a) (b)

(c)

38. (a) (b)

(c)

39. (a) (b)(c)

40. (a) (b)(c)

41.

42. Distances between the points:

43.

44. (a) (b)

45.

46. (a)

(b)

47. 48.

49. $3803.5 million 50. $2393.5 million

51.

52.

53.

54.

55. $3.31 per pound; 2001 56. 57.

58. (a) (b)

59. (a) The number of artists elected each year seems to benearly steady except for the first few years. From six toeight artists will be elected in 2008.

(b) The Rock and Roll Hall of Fame was opened in 1986.

60. (a) 1990s (b) 26.5%; 21.2% (c) $6.24

(d) Answers will vary.

61. 1998: $19,384.5 million; 2000: $20,223.0 million;2002: $21,061.5 million

62. (a) (b) 65

(c) No. There are many variables that will affect the finalexam score.

Math entrance test score

Fina

l exa

m s

core

2010

30

50

70

90

40

60

80

100

2010 30 50 7040 60 80

y

x

� 147%� 25%

� 250%� 91%

��5, 2�, ��7, 0�, ��3, 0�, ��5, �4��3, 6�, �2, 10�, �2, 4�, ��3, 4�(3, 3�, �1, 0�, �3, �3�, �5, 0��0, 1�, �4, 2�, �1, 4)

30�41 � 192 kilometers2�505 � 45 yards

��32, �9

4�, ��1, �32�, ��1

2, �34�

�74, �7

4�, �52, �3

2�, �134 , �5

4��x1 � 3x2

4,

y1 � 3y2

4 �� 3x1 � x2

4,

3y1 � y2

4 �, �x1 � x2

2,

y1 � y2

2 �,

�9, �3��7, 0��2xm � x1, 2ym � y1�

�29, �58, �29

��5 �2� ��45 �2

� ��50 �2

��5.6, 8.6��556.52

x

10

5

−5

15

20

5−5−10−15−20

(−16.8, 12.3)

(5.6, 4.9)

y

�1.25, 3.6��110.97

x

2

−2

4

6

8

2−2−4 4 6

(6.2, 5.4)

(−3.7, 1.8)

y

��14, � 5

12�

�2

6x

1

1

3

23

1

1

1

1

6

6

6

66

3

6

3

2

−

−

26

−

−

−−

− −,

− −,

( )

( )

y

��1, 76�

�82

3

x

( )( ), 1

−−−

25

134

222

3

12

52

12

12

32

52

,−

y

−2 −1

�6, 6�8�2

x

2

4

6

8

10

102 4 6 8

(2, 10)

(10, 2)

y

�2, 3�2�10

x

(5, 4)

(−1, 2)

1−1 2 3 4 5

3

−1

4

5

y

Precalculus with Limits, Answers to Section 1.1 2

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(Continued)

63. 64.

65.

66. 34 centimeters67. (a) (b)

(c)

68. (a)

(b)

(c)69. (a) (b) 2002

(c) Answers will vary. Sample answer: Technology nowenables us to transport information in many ways otherthan by mail. The Internet is one example.

70. (a)

(b) 1994(c) 2003; 42 teams

71.

(a) The point is reflected through the y-axis.(b) The point is reflected through the x-axis.(c) The point is reflected through the origin.

72. (a) First Set Second Set

Distance A to B 3Distance B to C 5Distance A to C 4

Right triangle Isosceles triangle

(b)

The first set of points is not collinear. The second set ofpoints is collinear.

(c) A set of three points is collinear if the sum of two distances among the points is exactly equal to the thirddistance.

73. False. The Midpoint Formula would be used 15 times.

74. True. Two sides of the triangle have lengths of andthe third side has a length of

75. No. It depends on the magnitudes of the quantities measured.

76. Use the Midpoint Formula to prove that the diagonals ofthe parallelogram bisect each other.

77. b 78. c

79. d 80. a

81. 82.

83. 84.

85. 86.

87. 88. or x ≥ �2x ≤ �1314 < x < 22

x ≤ �234x < 3

5

x ��3 ± �73

4x � 2 ± �11

x � 6x � 1

�a � b � 02

, c � 0

2 � � �a � b2

, c2�

�b � a2

, c � 0

2 � � �a � b2

, c2�

�18.�149,

y

x−2 2 4 6 8

−2

2

4

6

8

�40�10�10

x

(3, 5)

(2, 1)2

−4

−6

−2 2 4 6 8−4−6−8

−8

4

6

8

y

(−7, −3)

(−2, 1)

(−3, 5)

(7, −3)

y

x

Year (4 ↔ 1994)

Num

ber

of b

aske

tbal

l tea

ms

4 6 8 10 12

850

900

950

1000

1050 MenWomen

y

x6 7 8 9 10 11 12 13

Year (6 ↔ 1996)

Piec

es o

f m

ail (

in b

illio

ns)

180

185

190

195

200

205

210

h � 10 inches, w � 12.5 inches, l � 16 inchesw � 1.25h; V � �20 inches�h2

h

w

l

7.5 meters � 5 metersl � 1.5w; p � 5w

w

l

area � 800.64 square centimetersLength of side � 43 centimeters;

603.24�

� 48 feet3�4.47�

� 1.12 inches

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Section 1.2 (page 22)

Vocabulary Check (page 22)1. solution or solution point 2. graph3. intercepts 4. y-axis 5. circle; 6. numerical

1. (a) Yes (b) Yes 2. (a) Yes (b) No3. (a) No (b) Yes 4. (a) Yes (b) No5.

6.

7.

8.

9. x-intercepts: 10. x-intercept:

y-intercept: y-intercept:

11. intercept: 12. intercept:

intercept: intercept:


intercept:


intercept: intercept:

17. x-intercepts: 18. x-intercepts:

y-intercept: y-intercept:

19. intercept: 20. x-intercept:

intercepts: y-intercepts:

21. 22.

23. 24.

25. y-axis symmetry 26. x-axis symmetry

27. Origin symmetry 28. y-axis symmetry

29. Origin symmetry 30. y-axis symmetry

31. x-axis symmetry 32. Origin symmetry

–4 –3 –2 –1 1 2 3 4

–4

–3

–2

1

2

3

4

x

y

–4 –3 –2 1 2 3 4

–4

–3

–2

1

2

3

4

x

y

1 2 3 6 7 8

−4

−3

−2

−1

1

2

3

4

x

y

–4 –3 –1 1 3 4

1

2

−2

3

4

x

y

�0, ±1��0, ±�6 �y-

��1, 0��6, 0�x-

�0, �25��0, 0��±�5, 0��0, 0�, �2, 0��0, �10�y-�0, 7�y-

��10, 0�x-�73, 0�x-

�0, 2�y-

�12, 0�x-��4, 0�x-

�0, 8�y-�0, �6�y-

�83, 0�x-�6

5, 0�x-

�0, 9��0, 16��3, 0��±2, 0�

–4 –3 –1 1 3 4

–2

–1

1

2

3

6

4

x

y

1 2

3

4

5

4 5x

−2 −1−1

−2

y

–4 –3 –2 –1 2 3 4

–4

–3

–2

1

2

3

4

x

y

x541−1−2−3

4

7

2

1

−1

2

3

5

y

�h, k�; r


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x 0 1 2

y 7 5 3 1 0

�52, 0��2, 1��1, 3��0, 5��1, 7��x, y�

52�1

x 0 1 2 3

y 4 0 0

�3, 0��2, �2��1, �2��0, 0��1, 4��x, y�

�2�2

�1

x 0 1 2

y 0

�2, 12��43, 0��1, �1

4��0, �1��2, �52��x, y�

12�

14�1�

52

43�2

x 0 1 2

y 1 4 5 4 1

�2, 1��1, 4��0, 5��1, 4��2, 1��x, y�

�1�2

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(Continued)

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

Intercepts: Intercepts:

47.

Intercepts:

48.

Intercepts:

49. 50.

Intercept: Intercept:

51. 52.

Intercept: Intercepts:

53. 54.

Intercepts: Intercepts: �0, 0�, �6, 0��0, 0�, ��6, 0�

−10

−10

10

10−10

−10

10

10

��1, 0�, �0, 1��0, 0�

−10

−10

10

10−10

−10

10

10

�0, 4��0, 0�

−10

−10

10

10−10

−10

10

10

��2, 0�, �1, 0�, �0, �2�

−10

−10

10

10

�3, 0�, �1, 0�, �0, 3�

−10 10

−10

10

� 32, 0�, �0, �1��6, 0�, �0, 3�

−10

−10

10

10−10

−10

10

10

x21−1−2−3−4−6

4

−1

−4

−3

1

3

(−5, 0)

0, 5( )

0, − 5( )

y

–2 1 2 3 4

–3

–2

2

3

x

(0, −1)

(−1, 0) (0, 1)

y

x2 31−2−3

2

−1

−2

−3

1

3

(−1, 0) (1, 0)

(0, 1)

y

2−2

2

−2

4

6

8

10

12

4 6 8 10 12x

(6, 0)

(0, 6)

y

–4 –3 –2 –1 1 2–1

2

3

4

5

x(1, 0)

(0, 1)

y

1 2 3 4 5 6–1

1

2

3

4

5

x(3, 0)

y

–3 –2 –1 2 3

–4

–3

–2

1

2

x(1, 0)

(0, −1)

y

−4 −3 −2−1

1

2

4

5

6

7

1 2 3 4x

(0, 3)

−3, 03( (

y

x321−1−3−4−5

−6

−5

−4

−3

−2

2

−1

(−2, 0) (0, 0)

y

−1

2

3

4

−1 1 2 3 4x

−2

−2

(0, 0) (2, 0)

y

(0, −3)

32 , 0( )

y

x−1−2−3 1 2 3

−1

−2

−3

1

2

( (x

(0, 1) 13

y

−1−2−3−4 1 2 3 4−1

−2

−3

1

4

5

, 0

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(Continued)

55. 56.

Intercepts: Intercepts:

57. 58.

59.

60.

61.

62.

63. 64.

65. Center: Radius: 5 66. Center: Radius: 4

67. Center: Radius: 3 68. Center: Radius: 1

69. Center: Radius: 70. Center: Radius:

71. 72.

73. (a) (b) Answers will vary.

(c) (d)

(e) A regulation NFL playing field is 120 yards long andyards wide. The actual area is 6400 square yards.

74. (a) (b) Answers will vary.

(c) (d)

(e) Answers will vary.Sample answer: There are no fixed dimensions for a regulation major league soccer field, but they are generally 110 to 115 yards (100.6 to 105.2 meters) longand 75 yards (68.6 meters) wide. This makes the areaabout 8250 square yards (6901.2 square meters).

75. (a) and (b)

Year (20 ↔ 1920)

Lif

e ex

pect

ancy

20 40 60 80 100

100

80

60

40

20

t

y

x � 90, y � 90

0 1800

9000

y

x

5313

y � 86 23x � 86 2

3,

0 1800

8000

y

x

t1 65432

1600

3200

4800

6400

8000

Year

Dep

reci

ated

val

ue

y

250,000

200,000

150,000

100,000

50,000

1 2 3 4 5 6 7 8t

Year

Dep

reci

ated

val

ue

y

y

x−1 1 2 3 4

−1

−2

−3

−4

1

(2, −3)–1 1 2 3

1

3

x

y

( )12

, 12

43�2, �3�;3

2�12, 12�;

–2 –1 1 2

–1

1

3

x

y

(0, 1)

−3 −2 1

1

2 4 5x

−1

−2

−3

−4

−5

−6

−7

y

(1, −3)

�0, 1�;�1, �3�;

x

−2

−3

1

2

3

5

−5

531−1−2−3−5 2

y

(0, 0)

1−1−2

−2−3−4

−6

−3−4

1234

6

2 3 4 6x

y

(0, 0)

�0, 0�;�0, 0�;

x2 � y 2 � 17�x � 3� 2 � �y � 4� 2 � 25

�x � 3� 2 � �y � 2� 2 � 25

�x � 1� 2 � �y � 2� 2 � 5

�x � 7�2 � �y � 4�2 � 49

�x � 2� 2 � �y � 1� 2 � 16

x2 � y 2 � 25x2 � y 2 � 16

�±2, 0�, �0, 2��3, 0�, �0, 3�

−10

−10

10

10−10

−10

10

10


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333202CB01a_AN.qxd 4/26/06 6:30 PM


(Continued)

(c) 66.0 years (d) 2005: 77.0 years; 2010: 77.1 years(e) Answers will vary.

76. (a)

(b)

When the resistance is 1.1 ohms.

(c) Answers will vary.

(d) As the diameter of the copper wire increases, theresistance decreases.

77. False. A graph is symmetric with respect to the x-axis if,whenever is on the graph is also on thegraph.

78. True. It is possible for a graph to have no intercepts, oneintercept, or several intercepts, as shown in Figure 1.5.

79. The viewing window is incorrect. Change the viewing window. Answers will vary.

80. (a) (b)

81. 82. 83. 84.

85. 86. 87. 88. 6�y3��t�5�2�5 � 3�10�7xx

�x� 4�x2�2x�749x5, 4x3, �7

a � 0, b � 1a � 1, b � 0

�x, �y��x, y�

x � 85.5,

y

x20 40 60 80 100

50100150200250300350400450

Res

ista

nce

(in

ohm

s)

Diameter of wire (in mils)

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x 5 10 20 30 40 50

y 430.43 107.33 26.56 11.60 6.36 3.94

x 60 70 80 90 100

y 2.62 1.83 1.31 0.96 0.71

333202CB01a_AN.qxd 4/26/06 6:30 PM


Vocabulary Check (page 34)1. linear 2. slope 3. parallel4. perpendicular 5. rate or rate of change6. linear extrapolation7. a. iii b. i c. v d. ii e. iv

1. (a) (b) (c)2. (a) (b) (c)3. 4.

5. 6. 7.8. 9. 10.

-intercept: -intercept:

11. 12.-intercept: -intercept:

13. is undefined. 14.There is no -intercept. -intercept:



19. m is undefined. 20. m is undefined.There is no y-intercept. There is no y-intercept.

21. 22.

23. 24.

m is undefined. m � �52

−6 −2 2 4

−10

−4

−2

x(−4, 0)

(0, −10)

y

–8 –2

–2

2

4

6

x

(−6, −1)

(−6, 4)

y

m � �4m � 2

−2 2 4 6

−4

−2

2

4

x

(2, 4)

(4, −4)

y

–5 –4 –3 –1 1 2 3

1

2

4

5

6

x

(−3, −2)

(1, 6)

y

y

x−4

−4

−3

−2

−1

1

2

3

4

−3 −2 −1 1 3 4−4−6−7 −3 −2 −1

3

2

4

−2

−3

−1

1

−4

1

y

x

y

x−4

−6

−5

−3

−2

−1

1

2

−3 −2 −1 1 2 3 4

(0, −4)

x

(0, 3)

−3 −2 −1−1

1

1

2

4

5

2 3

y

�0, �4�y�0, 3�ym � 0;m � 0;

−1 1 2 3 4

1

2

4

5

x

(0, 3)

y

x

3

2 31−1

4

5 (0, 5)

2

1

4 6 7

−2

y

�0, 3�y�0, 5�ym � �

23;m � �

76;

x

53( )

−1−2 21

−1

−2

−3

1

0, −

y

–1 1 2 3

–2

–1

1

2

x

y

�0, �53�yy

m � 0;m

x

y

1−1 2

1

2

3

4

5

6

3 4 5 6 7

(0, 6)

x

y

−1

1

−2

1 32

1

2

3

5

6

7

4 5 6 7 8

(0, 4)

�0, 6�y�0, 4�ym � �

32;m � �

12;

–2 2 4 6 10 12

–10

–8

–6

–4

2

x

(0, −10)

y

x

3

2−1−2−3−4 31

4

5

(0, 3)

y

�0, �10�y�0, 3�ym � 1;m � 5;�1�45

232

x

12

(−4, 1)

−6 −2

−2

2

4

undefined

m = −3m = 3

m =

y

x1 2

1

2

(2, 3)

m = 1

m = 2

m = 0

m = −3

y

L3L1L2

L1L3L2


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(Continued)

25. 26.

27. 28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. 52.

−6 −4 −2 2 4 6

−4

2

4

6

x

(−4, −4)

(4, 3)

y

–6 –4 –2 2 6

–4

–2

4

6

8

x

(5, −1)

(−5, 5)

y

y �78 x �

12y � �

35 x � 2

x−2−4 2 4

−8

−4

−6

2

(2.3, −8.5)

−10

6 8

y

x−1 1−2−3−4−6−7

1

2

3

−2

−3

−4

−5

(−5.1, 1.8)

y

y � �2.5x � 2.75y � 5x � 27.3

y

x

, 32

12

−( (

−1−2−3 1 2 3−1

−2

1

2

3

4

x

y

1 2

1

2

3

4

5

−1−1

3 4 5

4, 52))

y �32y �

52

y

x−12 −8 −6 −4 −2 2

−6

−4

−2

2

4

6

8

(−10, 4)

–4 –2 2 4

–6

–4

–2

2

4

6

x(6, −1)

y

x � �10x � 6

−2 2

−2

x

(−2, −5)

y

–1 1 2 3 4

–2

–1

1

2

3

4

x(4, 0)

y

y �34 x �

72y � �

13 x �

43

x

3

2−1−2−3 31

4

5

(0, 0)

2

1

y

–6 –4 –2 2 4 6

–6

–4

–2

4

6

x

(−3, 6)

y

y � 4xy � �2x

5

5

10

x

(0, 10)

y

–2 –1 1 2 3 4

–2

–1

1

2

x

(0, −2)

y

y � �x � 10y � 3x � 2

�5, �9��1, �7�,��3, �5�,�9, �1�, �11, 0�, �13, 1�

�3, �15��1, �11�,��2, �5�,��4, 6�, ��3, 8�, ��2, 10�

�0, �1��2, �1�,��4, �1�,��8, 0�, ��8, 2�, ��8, 3�

�11, �7��9, �5�,�0, 4�,�6, �5�, �7, �4�, �8, �3�

��4, 5��4, 3�,��4, 0�,�0, 1�, �3, 1�, ��1, 1�

m � 1.425m � 0.15

x1 2 3 4 5−1−2−3−4−5

1

−2

−3

−4

−5

−6

−8

−9

(−1.75, −8.3)

(2.25, −2.6)

y

x

6

62−2−4−6−2

−4

(−5.2, 1.6)(4.8, 3.1)4

8

4

y

m � �83m � �

17

x14

1

54

14

14

12

34

12

34

14

−

12

−( ( , −

78

34( ( ,

y

x4 5

1

2

( (32

13

− , −

( (112

43

, −

6

3

−1

−2

−3

−4

−5

−6

y

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333202CB01a_AN.qxd 4/26/06 6:30 PM

(Continued)

53. 54.

55. 56.

57. 58.

59. 60.

61. 62.

63. 64.

65. Perpendicular 66. Neither parallel nor perpendicular

67. Parallel 68. Perpendicular

69. (a) (b)

70. (a) (b)

71. (a) (b)

72. (a) (b)

73. (a) (b)

74. (a) (b)

75. (a) (b)

76. (a) (b)

77. (a) (b)

78. (a) (b)

79. 80.

81. 82.

83. 84.

85. Line (b) is perpendicular to line (c).

86. Line (a) is parallel to line (c). Line (b) is perpendicular toline (a) and line (c).

87. Line (a) is parallel to line (b).Line (c) is perpendicular to line (a) and line (b).

−10 14

−8

(b)

(a)

8 (c)

−9 9

−6

(c)

(b)

(a)

6

−6 6

−4

(c)

4(b) (a)

x � y � 1 � 0x � y � 3 � 0

3x � y � 2 � 012x � 3y � 2 � 0

4x � 3y � 12 � 03x � 2y � 6 � 0

y �13 x � 0.1y � �3x � 13.1

y � �x � 9.3y � x � 4.3

y � 1x � �5

y � 5x � 2

x � 4y � �2

x � �1y � 0

y �35 x �

940y � �

53 x �

5324

y �43 x �

12772y � �

34 x �

38

y � x � 5y � �x � 1

y � �12 x � 2y � 2x � 3

y

x−3

−3

−2

−1

1

2

3

−2 −1 1 2 3

(1.5, 0.2)

(1.5, −2)

x

y

73 )) , 1

73 )) , −8

8642 3 5 71−1

2

1

−2

−3

−4

−5

−6

−7

−8

x � 1.5x �73

y

x

(−6, −2) , −215( (

−1−2−3−4−5−6 1 2−1

−3

−4

−5

1

2

3

y

x1−1 2

1

−2

−3

2

3

3 4 5

13, −1)) (2, −1)

y � �2y � �1

−8−10 −6

−8

−6

−4

2

4

6

x

(−8, 0.6)

(2, −2.4)

y

1−3 2 3

1

2

3

−3

−2

x

(1, 0.6)

(−2, −0.6)

y

y � �0.3x � 1.8y � 0.4x � 0.2

x1 2

−1

1

2

3

32( (,3

4

− 74( (,4

3

−1−2

y

x21

1

2

−2( (, −

−1−2

y

910

95

( (, −− 110

35

y � �325 x �

159100y � �

65 x �

1825

x

23( )6, −

1 2 3 4

−2

−1

1

2

3

4

(1, 1)

y

x

12

12

54( (

( )

−1 2 31

1

−1

,

2,

2

3

y

y � �13 x �

43y � �

12 x �

32

−2 2 4 6 8

−2

2

6

8

x

(−1, 4) (6, 4)

y

–10 –6 –4 –2

–2

2

4

6

8

x(−8, 1)

(−8, 7)

y

y � 4x � �8


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333202CB01a_AN.qxd 4/26/06 6:30 PM


(Continued)

88. Line (a) is parallel to line (b). Line (c) is perpendicular toline (a) and line (b).

89. 90.91. 92.93. (a) Sales increasing 135 units per year

(b) No change in sales(c) Sales decreasing 40 units per year

94. (a) Revenues increasing $400 per day.(b) Revenues increasing $100 per day.(c) No change in revenues.

95. (a) Salary increased greatest from 1990 to 1992; Leastfrom 1992 to 1994

(b) Slope of line from 1990 to 2002 is about 2351.83(c) Salary increased an average of $2351.83 over the 12

years between 1990 and 200296. (a) Net profit showed largest increase in years

2002–2003; Least increase in years 2000–2001(b) Slope of line connecting 1994 and 2003 is about 9.18(c) Annual profits increased a factor of about 9.18 over

those years97. 12 feet98. (a) and (b)

(c)(d) For every 12 horizontal measurements, the vertical

measurement decreases by 1.(e) “8.3% grade”

99. 100.101. b; The slope is which represents the decrease in the

amount of the loan each week. The y-intercept is which represents the original amount of the loan.

102. c; The slope is 2, which represents the hourly wage perunit produced. The y-intercept is which repre-sents the initial hourly wage.

103. a; The slope is 0.32, which represents the increase intravel cost for each mile driven. The y-intercept is which represents the amount per day for food.

104. d; The slope is which represents the decrease in thevalue of the word processor each year. The y-intercept is

which represents the initial purchase price of thecomputer.

105.

106.

Answers will vary.

107. 108.

109. (a)

(b) (c)

110. (a) Average annual change from 1990 to 2003 was 934.

(b) 1994; 40,267; 1998: 44,003; 2002: 47,739

(c)

(d) Answers will vary.

111. 112.

113. (a) (b)

(c) (d)

114. (a) (b) 45 (c) 49

115. (a) (b)

(c) (d) 8 meters

116. 117.

118. Answers will vary. Sample answer:

119. (a) and (b)

(c) Answers will vary. Sample answer:y � 11.72x � 14.1

y

x2 4 6 8 10 12

150

125

100

25

50

75

Year (0 ↔ 1990)

Cel

lula

r ph

one

subs

crib

ers

(in

mill

ions

)

y � 327.3t � 1854

C � 0.38x � 120W � 0.07S � 2500

m � 8,

0100

150

y � 8x � 50

15 m

10 m

x

x

x � �1

15 p �2663

t � 3561 hoursP � 10.25t � 36,500

R � 27tC � 16.75t � 36,500

W � 0.75x � 11.50S � 0.85L

m � 934y�t� � 934t � 36,531;

m � 179.5y�10� � 42,366y�8� � 42,007;

y�t� � 179.5t � 40,571

V � 25,000 � 2300tV � �175t � 875

y�20� � �4168.5;

y�18� � �2669.5;y�t� � �749.5t � 10,821.5;

y�20� � $7.42y�18� � $6.45;y � 0.4825t � 2.2325;

�0, 750�

�100,

�0, 30�

�0, 8.50�

�0, 200��20,

V�t� � 4.5t � 133.5V�t� � 3165 � 125t

y � �112 x

Ver

tical

mea

sure

men

ts

Horizontal measurements

y

x600 1200 1800 2400

−50

−100

−150

−200

128x � 168y � 39 � 080x � 12y � 139 � 05x � 13y � 2 � 03x � 2y � 1 � 0

−14 16

−10

(b)

(c)

(a)

10

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333202CB01a_AN.qxd 4/26/06 6:30 PM

(Continued)

(d) Answers will vary. Sample answer: The y-interceptindicates that initially there were million sub-scribers which doesn’t make sense in the context ofthis problem. Each year, the number of cellular phonesubscribers increases by 11.72 million.

(e) The model is accurate.(f ) Answers will vary. Sample answer: 196.9 million

120. (a) and (b)

(c) (d) 87

(e) Vertical shift four units upward

121. False. The slope with the greatest magnitude correspondsto the steepest line.

122. False. The slope of the first line is and the slope of thesecond line is

123. Find the distance between each two points and use thePythagorean Theorem.

124. The slope of a vertical line is undefined because divisionby zero is undefined.

125. No. The slope cannot be determined without knowing thescale on the -axis. The slopes could be the same.

126. The line with a slope of is steeper. The slope with thegreatest magnitude corresponds to the steepest line.

127. -intercept: initial cost; Slope: annual depreciation

128. No. The slopes of two perpendicular lines have oppositesigns (assume that neither line is vertical or horizontal).

129. d 130. c 131. a

132. b 133. 134.

135. 136. 137. No solution

138. 139. Answers will vary.19, 25

4 ± �1372, 7

52�1

V

�4

y

�117 .

27

y � 4x � 19

x10 12 20181614

50

60

70

80

90

100

y

Average quiz score

Ave

rage

test

sco

re

�14.1


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333202CB01a_AN.qxd 4/26/06 6:30 PM



Vocabulary Check (page 48)1. domain; range; function2. verbally; numerically; graphically; algebraically3. independent; dependent 4. piecewise-defined5. implied domain 6. difference quotient

1. Yes 2. No 3. No 4. Yes5. Yes, each input value has exactly one output value.6. No, the input values of 0 and 1 each have two different

output values.7. No, the input values of 7 and 10 each have two different

output values.8. Yes, it does not matter that each input value has the same

output value.9. (a) Function

(b) Not a function, because the element 1 in A corre-sponds to two elements, and 1, in B.

(c) Function(d) Not a function, because not every element in A is

matched with an element in B.10. (a) Not a function, because the element in A corre-

sponds to two elements, 2 and 3, in B.(b) Function(c) Not a function from A to B. (It is instead a function

from B to A.)(d) Function

11. Each is a function. For each year there corresponds oneand only one circulation.

12. 11 million 13. Not a function 14. Not a function15. Function 16. Not a function 17. Function18. Not a function 19. Not a function 20. Function21. Function 22. Not a function 23. Not a function24. Function 25. (a) (b) (c)26. (a) 7 (b) 0 (c)27. (a) (b) (c)28. (a) 0 (b) (c)29. (a) 1 (b) 2.5 (c)30. (a) 2 (b) 5 (c)

31. (a) (b) Undefined (c)

32. (a) (b) Undefined (c)

33. (a) 1 (b) (c)

34. (a) 6 (b) 6 (c)35. (a) (b) 2 (c) 636. (a) 6 (b) 3 (c) 1037. (a) (b) 4 (c) 9

38. (a) 19 (b) 17 (c) 039.

40.

41.

42.

43.

44.

45. 5 46. 47. 48. 49.50. 3, 5 51. 52. 1, 53.54. 0, 55. 3, 0 56. 457. All real numbers 58. All real numbers59. All real numbers t except 60. All real numbers y except 61. All real numbers y such that 62. All real numbers63. All real numbers x such that 64. All real numbers x such that and 65. All real numbers x except 66. All real numbers x except 67. All real numbers s such that except 68. All real numbers x such that 69. All real numbers x such that 70. All real numbers x such that

71.72.73.74.

75. 76.

77. 78.

79. 80.

81.

82. 8x � 4h � 2, h � 0

3x 2 � 3xh � h2 � 3, h � 0

��5 � h�, h � 03 � h, h � 0

h�x� � c��x�; c � 3r�x� �cx; c � 32

f �x� � cx; c �14g�x� � cx2; c � �2

��2, 1�, ��1, 0�, �0, 1�, �1, 2�, �2, 3��2, 4�, ��1, 3�, �0, 2�, �1, 3�, �2, 4��2, 1�, ��1, �2�, �0, �3�, �1, �2�, �2, 1��2, 4�, ��1, 1�, �0, 0�, �1, 1�, �2, 4�

x > 3x < �3,x > 0x > �6

s � 4s ≥ 1x � 0, 2x � 0, �2

x ≥ 0x ≤ �3�1 ≤ x ≤ 1

y ≥ 0y � �5t � 0

±22, �1±20, ±1

±3±2�343�

15

�7

�1x 2 � 4

�x � 1�x � 1

�1

2x 2 � 3

x 2

11

4

1y2 � 6y

�19

�x � 23 � 2�x�x2 � 2x�0.75

323 �r39

2�36�

1 � 3s2x � 5�9�1

c

�2

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x 0 1 2

5 4 1 092f �x�

�1�2

x 0 1 2

1 1�2�3�2f �x�

�1�2

t

1 0 112

12h �t�

�1�2�3�4�5

3 4 5 6 7

0 1 2�3�2g�x�

x

0 1 4

1 1�1�1�1f �s�

52

32s

x 1 2 3 4 5

8 5 0 1 2f �x�

333202CB01a_AN.qxd 4/26/06 6:30 PM

(Continued)

83. 84. 85.

86. 87. 88.

89. (a) The maximum volume is 1024 cubic centimeters.(b)

Yes, V is a function of x.(c)

90. (a) The maximum profit is $3375.(b)

Yes, is a function of (c)

91.

92.

93. Yes, the ball will be at a height of 6 feet.

94. 1991: $42 billion 95. 1990: $27,3001992: $47 billion 1991: $28,0521993: $52 billion 1992: $29,1681994: $57 billion 1993: $30,6481995: $62 billion 1994: $32,4921996: $67 billion 1995: $34,7001997: $72 billion 1996: $37,2721998: $85.6 billion 1997: $40,2081999: $104.3 billion 1998: $41,3002000: $123 billion 1999: $43,8002001: $141.7 billion 2000: $46,3002002: $160.4 billion 2001: $48,800

2002: $51,300

96. (a)

(b)

(c) 18 inches 18 inches 36 inches97. (a) (b)

(c)

98. (a) (b)

99. (a)

(b)

The revenue is maximum when 120 people take the trip.100. (a)

The deeper the water, the greater the force.(b) 21 feet (c) 21.37 feet

101. (a)

(b)102. (a) 1.428. Approximately 1.428 species of fish became

threatened and/or endangered each year between 1996and 2003.

(b)

(c)

(d) The algebraic model is a good fit to the actual data.(e) The model found in part (b) is a

better model than the model given by the graphingutility.

N � 1.5x � 107;

N � �2x � 104,2x � 103,126,

6 ≤ x ≤ 78 ≤ x ≤ 1112 ≤ x ≤ 13

d ≥ 3000h � �d2 � 30002,

3000 ft

dh

n ≥ 80R �240n � n2

20,

C �C

x�

6000

x� 0.95C � 6000 � 0.95x

P � 5.68x � 98,000R � 17.98xC � 12.30x � 98,000

��

00 30

12,000

0 < x < 27 � 108x2 � 4x 3, V � x2y � x2�108 � 4x�

0 < x < 6A � 2xy � 2x�36 � x 2,

x > 2A �x2

2�x � 2�,

x > 100P � 45x � 0.15x2,x.P

x110 130 150 170

3150

3100

3200

3250

3300

3350

3400

Order size

Prof

it

P

0 < x < 12V � x�24 � 2x�2,

x1 2 6543

200

400

600

800

1000

1200

Height

Vol

ume

V

A �C 2

4�A �

P2

16x2�3 � 4

x � 8

�5x � 5x � 5

1t � 2

, t � 1�x � 3

9x2 , x � 3


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n 90 100 110 120 130 140 150

$675 $700 $715 $720 $715 $700 $675R�n�

5 10 20

26,474.08 149,760.00 847,170.49F�y�

y

30 40

2,334,527.36 4,792,320.00F�y�

y

x 6 7 8 9 10 11 12 13

N 116 118 119 121 123 125 126 126

333202CB01a_AN.qxd 4/26/06 6:30 PM


(Continued)

103. False. The range is 104. True. Each -value corresponds to one -value.105. The domain is the set of inputs of the function, and the

range is the set of outputs.106. Domain of all real numbers x such that

Domain of all real numbers x107. (a) Yes. The amount you pay in sales tax will increase as

the price of the item purchased increases.(b) No. The length of time that you study will not neces-

sarily determine how well you do on an exam.

108. (a) No, not necessarily. It depends on how you manageyour money.

(b) Yes. The speed of the baseball will increase as theheight from which it was dropped increases.

109. 110. 8

111. 112.

113. 114.

115. 116. 10x � 18y � 49 � 010x � 9y � 15 � 0

x � y � 10 � 02x � 3y � 11 � 0

23�

15

158

g�x�:x ≥ �2f �x�:

yx �1, ��.

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333202CB01a_AN.qxd 4/26/06 6:30 PM


Vocabulary Check (page 61)1. ordered pairs 2. Vertical Line Test3. zeros 4. decreasing5. maximum 6. average rate of change; secant7. odd 8. even

1. Domain: 2. Domain: Range: Range:

3. Domain: 4. Domain: Range: Range:

5. (a) 0 (b) (c) 0 (d) 6. (a) 4 (b) 4 (c) 2 (d) 07. (a) (b) 0 (c) 1 (d)8. (a) 0 (b) 1 (c) 2 (d) 39. Function 10. Function 11. Not a function

12. Not a function 13. Function 14. Not a function15. , 6 16. 17. 0 18. 2, 7

19. 20. 21. 22. 0,

23. 24.25. 26.

0, 727. 28.

2629. 30.

31. Increasing on 32. Increasing on Decreasing on 33. Increasing on and

Decreasing on 34. Increasing on Decreasing on 35. Increasing on and Constant on 36. Increasing on Decreasing on 37. Increasing on Decreasing on

Constant on 38. Increasing on

Decreasing on

39. 40.

Constant on Increasing on

41. 42.

Decreasing on Increasing on

Increasing on Decreasing on

43. Increasing on

Decreasing on

44. Increasing on

Decreasing on

45. 46.

Decreasing on Increasing on

Decreasing on

47. 48.

Increasing on Decreasing on

Increasing on

49. Relative minimum: �1, �9�−8 10

−10

2

�0, ��, 0��0, ��

−6 6

−2

6

0 60

4

��3, �2��2, ��, 1�

−9 9

−3

9

−4 2

−1

3

��, �1�, �0, 1�

��1, 0�, �1, ��

−6 6

−4

4

�0, ��, 0�

−3 3

−3

1

��, 0��0, ��0, ��, 0�

−4 5

−5

1

−6 6

−1

7

��, ��, ��

−3 3

−2

2

−3 30

4

��2, �1�, ��1, 0��, �2�, �0, ��

��1, 1�

��, �1��1, ��;��1, 0��, �1�, �0, ��;

�0, 2��2, ��;��, 0��, �1��1, ��;

�0, 2��2, ��, 0�

��, 2��2, ��;��, ��

±3�2

213

−30

−15 25

10

3−3

−2

2

�112

−12

−4 28

4

3−6

−1

5

�53

−14

−2 13

3

9−9

−6

6

�23

12

±53±1

2, 6±3, 40, ±�2

23�8,�

52

�3�3

�2�1�1, 1 0, 4�

��, 1�, �1, �� 4, 4� 0, �� 0, ��

��, ��, �1�, 1, ��


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333202CB01a_AN.qxd 4/26/06 6:30 PM


(Continued)

50.

Relative minimum:

51.

Relative maximum:

52.

Relative maximum:

53.

Relative maximum: Relative minimum:

54.

Relative maximum:Relative minimum:

55. 56.

57. 58.

59. 60.

61. 62.

for all

63. The average rate of change from to is 64. The average rate of change from to is 3.65. The average rate of change from to is 18.66. The average rate of change from to is 4.67. The average rate of change from to is 0.68. The average rate of change from to is 0.69. The average rate of change from to is 70. The average rate of change from to is 71. Even; -axis symmetry72. Neither even nor odd; no symmetry73. Odd; origin symmetry 74. Odd; origin symmetry75. Neither even nor odd; no symmetry76. Even; -axis symmetry77. 78.79. 80.81. 82. 83.

84. L �2y

L � 4 � y 2L � 2 � 3�2yL �12y 2

h � 2 � 3�xh � 2x � x2

h � 3 � 4x � x2h � �x 2 � 4x � 3y

y�

15.x2 � 8x1 � 3

�14.x2 � 11x1 � 3

x2 � 6x1 � 1x2 � 3x1 � 1x2 � 5x1 � 1x2 � 5x1 � 1x2 � 3x1 � 0

�2.x2 � 3x1 � 0

��, ��xf �x� < 0

−2 −1 1 2

2

3

4

x

y

–2 –1 1 2

–4

–3

–2

x

y

�2, �� 1, ��−2 −1 1 2

2

3

4

x

yy

x

−1−1 1 2 3 4 5

1

2

3

4

5

��, 0�, 4, ��, �1�, 0, ��

1 2 3

−4

−3

−2

−1

x

yy

x−3 −2 −1 1 2

−1

1

2

3

4

5

3

�12, ��, 4�

−2 −1 1 2

2

3

4

x

y

–1 1 2 3 4 5

1

−1

2

3

4

5

x

y

�2.15, �5.08��0.15, 1.08�

−7 8

−7

3

�1.12, �4.06��1.79, 8.21�

−12 12

−6

10

�2.25, 10.125�−4

12−12

12

�1.5, 0.25�−4

6−3

2

�13, �16

3 �

−7 8

−7

3

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333202CB01a_AN.qxd 4/26/06 6:30 PM

(Continued)

85. (a) (b) 30 watts

86. (a)

(b) The model fits the data very well.(c) The temperature was increasing from 6 A.M. to

noon and again from 2 A.M. to 6 A.M.The temperature was decreasing from noon

to 2 A.M.(d) The maximum temperature was 63.93°F and the mini-

mum temperature was 33.98°F.(e) Answers will vary.

87. (a) Ten thousands (b) Ten millions (c) Percents88. (a)

(b)

(c) Square with sides of meters

89. (a)

(b) The average rate of change from 2002 to 2007 is408.56. The estimated revenue is increasing each yearat a fast pace.

90. (a)

(b) The average rate of change of the model from 1992 to2002 is 14.73. This shows that the number of studentsenrolling in college increased at a fairly steady rate.

(c) The five-year time period when the rate of change wasthe least was from 1992 to 1997 with a 6.05 averagerate, and the greatest from 1997 to 2002 with a 23.41average rate.

91. (a)

(b)

(c) Average rate of change

(d) The slope of the secant line is positive.

(e) Secant line:

(f )

92. (a)

(b)


(d) The slope of the secant line is positive.

(e)

(f)

93. (a)

(b)


(d) The slope of the secant line is negative.

(e) Secant line:

(f )

080

270

�8t � 240

� �8

080

270

s � �16t2 � 120t

050

100

Secant line � 8t � 6.5

� 8

050

100

s � �16t2 � 72t � 6.5

050

100

16t � 6

� 16

050

100

s � �16t2 � 64t � 6

400122

600

072

2200

4�2

32 ≤ A ≤ 64

0 40

80

0 ≤ x ≤ 4A � 64 � 2x2,

�x � 24�.�x � 20��x � 6�,

�x � 0�

0240

70

09020

6000


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333202CB01a_AN.qxd 4/26/06 6:31 PM


(Continued)

94. (a)(b)

(c) Average rate of change(d) The slope of the secant line is negative.(e)(f)

95. (a)(b)

(c) Average rate of change(d) The slope of the secant line is negative.(e) Secant line: (f )

96. (a)(b)

(c) Average rate of change(d) The slope of the secant line is negative.(e)(f)

97. False. The function has a domain of allreal numbers.

98. False. An odd function is symmetric with respect to the origin, so its domain must include negative values.

99. (a) Even. The graph is a reflection in the -axis.(b) Even. The graph is a reflection in the -axis.(c) Even. The graph is a vertical translation of f.(d) Neither. The graph is a horizontal translation of f.

100. Yes. For each value of there corresponds one and onlyone value of

101. (a) (b)

102. (a) (b)

103. (a) (b)

104. (a) (b)

105. (a) (b)

(c) (d)

(e) (f)

All the graphs pass through the origin. The graphs of theodd powers of are symmetric with respect to the origin,and the graphs of the even powers are symmetric withrespect to the -axis. As the powers increase, the graphsbecome flatter in the interval

106. Both graphs will pass through the origin. will besymmetric with respect to the origin, and will besymmetric with respect to the -axis.

107. 0, 10 108. 15 109. 0, 110.

111. (a) 37 (b) (c)

112. (a) (b) 144 (c)

113. (a) (b)

(c) The given value is not in the domain of the function.

114. (a) (b) (c)

115. 116. h � 0�6 � h,h � 0h � 4,

139 � 2�3�8716�3

2�7 � 9�9

x2 � 18x � 56�24

5x � 43�28

54±1�5,

−6 6

−4

4

−6 6

−4

4

yy � x 8y � x 7

�1 < x < 1.y

x

−6 6

−4

4

−6 6

−4

4

−6 6

−4

4

−6 6

−4

4

−6 6

−4

4

−6 6

−4

4

��5, 1�(�5, �1��4, �9��4, 9��5

3, 7��53, �7�

�32, �4��3

2, 4�x.

y

yx

f �x� � �x2 � 1

030

120

Secant line � �48t � 112

� �48

030

120

s � �16t2 � 80

040

140

�32t � 120

� �32

040

140

s � �16t2 � 120

060

175

Secant line � �16t � 160

� �16

060

175

s � �16t2 � 96t

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333202CB01a_AN.qxd 4/26/06 6:31 PM


Vocabulary Check (page 71)1. g 2. i 3. h 4. a 5. b6. e 7. f 8. c 9. d

1. 2.(a) (a)

(b) (b)

3. 4.(a) (a)

(b) (b)

5. 6.(a) (a)

(b) (b)

7. 8.(a) (a)

(b) (b)

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

−2

−4 11

8

10−5

−5

5

−5

−4 11

5

9−1

−1

12

−3

−6 3

3

8−7

−3

7

−9

−6 6

15

6−6

−4

4

−28

−4 12

6

20− 10

−2

18

−6

−6 12

20

6−6

−4

4

−4

−6 6

4

6−6

−6

2

−4

−6 6

4

6−6

−4

4

−14

−12

−8

−6

−4

−2−2

2

2 4 6 8 12 14

y

x

y

x

−9

−8

−6

−5

−4

−3

−2

−1

1

1 2 3 4 5 6 8 9

f �x� �34 x � 8f �x� �

67 x �

457

y

x−2

2

2 4 6 8 10 12 14

4

6

10

12

14

16

y

x−3 −2 −1 1 2

−3

−2

1

2

3

3

f �x� � �12 x � 7f �x� � �1

y

x−5

−7−6−5−4−3−2

1

2

−4 −3 −2 −1 2 3 4 5

y

x

2

2 6 8 10 12

4

6

8

10

12

f �x� � 5x � 6f �x� � �3x � 11

y

x−1−2−3−4 1 2 3 4

−1

1

2

3

4

5

y

x−1 1 2

1

2

3

4

5

6

3 4 5 6 7

f �x� �52 x �

12f �x� � �2x � 6


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333202CB01a_AN.qxd 4/26/06 6:31 PM


(Continued)

25. 26.

27. 28.

29. (a) 2 (b) 2 (c) (d) 3

30. (a) (b) 0 (c) 18 (d) 6

31. (a) 1 (b) 3 (c) 7 (d)

32. (a) 7 (b) (c) 31 (d) 11

33. (a) 6 (b) (c) 6 (d)

34. (a) 8 (b) 2 (c) 6 (d) 13

35. (a) (b) (c) (d) 41

36. (a) (b) (c) 6 (d)

37. 38.

39. 40.

41. 42.

43. 44.

45. 46.

47. 48.

49. 50.

51. (a) (b) Domain: ;

Range:

(c) Sawtooth pattern

52. (a) (b) Domain: ;

Range:

(c) Sawtooth pattern

53. (a) (b)

54. (a) (b)

55. (a) (b)

56. (a) (b) g�x� �1x

� 2f �x� �1x

g�x� � �x � 1�3 � 2f �x� � x3

g�x� � 1 � �x � 2f �x� � �x

g�x� � �x � 2� � 1f �x� � �x�

0, 2�

��, ��

−9 9

−4

8

0, 2�

��, ��

−9

−4

9

8

−3 −2 −1 1

−3

−2

−1

2

1

3

32

y

x

y

x−1−3−4 1 2 3 4

−1

−2

−3

1

2

3

4

5

y

x−1−3−4 1 2 3 4

−1

−2

1

2

4

5

6

–4 –2 2 4 6 8–2

2

4

8

10

x

y

−1 1 2 3 4 5

1

2

3

4

x

y

–4 –3 –2 –1 1 2 3 4

–3

–2

1

3

4

5

x

y

y

x

−16

−14

−12

−10

−10 −4 −2 2 8 10

−8

−6

−4

2

4

–1 1 2 3 4

–2

–1

1

3

4

x

y

y

x−3 −2 −1−4 1 2 3 4

−2

−1

−3

−6

1

2

y

x−3−4 1 2 3 4

−2

−3

−4

1

2

3

4

y

x−6

−6

−4

2

4

6

−4 −2 1 2 4 6

x1 2

2

−2

−5

−6

43−1−2−3−4

1

y

y

x−2−3−4 1 2 3 4

−12

−16

4

8

12

16

y

x−4 −3 −2 −1 3

4

2

3

−2

−1

−3

−4

4

�29�85�22

�1�4�10

�22�11

�1

�19

�6

�4

−5

−3 12

5

3−9

−4

4

−2

−9 9

10

6−6

−4

4

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333202CB01a_AN.qxd 4/26/06 6:31 PM

(Continued)

57. (a) (b)

58. (a) (b)

59. (a) (b)

60. (a) (b)

61. (a) (b) $5.64

62. (a) is the appropriate model, because the cost does notincrease until after the next minute of conversation hasstarted.

(b)

$7.89

63. (a) (b) $50.25

64. (a)(b)

65. (a)

(b)

66.

Total accumulation

67. (a)

Answers will vary. Sample answer: The domain isdetermined by inspection of a graph of the data with thetwo models.

(b)

(c) These values representthe revenue for the months of May and November,respectively.

(d) These values are quite close to the actual data values.68. Interval Input Pipe Drainpipe 1 Drainpipe 2

Open Closed ClosedOpen Open ClosedClosed Closed ClosedClosed Closed OpenOpen Open OpenOpen Closed OpenOpen Open OpenOpen Open Closed

69. False. A piecewise-defined function is a function that isdefined by two or more equations over a specified domain.That domain may or may not include - and -intercepts.

70. True. The solution sets are the same.

71.

72. 73.

74.75. Neither76. Neither

543210− 1

x25

x < 52

3−1−2−3

x

210

x ≤ 1f �x� � �x2,

7 � x, x ≤ 2

x > 2

f �x� � ��43x � 6,

�25x �

165 ,

0 ≤ x ≤ 3

3 < x ≤ 8

yx

50, 60� 45, 50� 40, 45� 30, 40� 20, 30� 10, 20� 5, 10� 0, 5�

f �5� � 11.575, f �11� � 4.63;

Month (1 ↔ January)

Rev

enue

(in

thou

sand

s of

dol

lars

) 2018161412108642

y

x1 2 3 4 5 6 7 8 9 10 11 12

1 ≤ x ≤ 6

6 < x ≤ 12f �x� � �0.505x2 �

�1.97x �

1.47x

26.3,

� 6.3,� 14.5 inches

y

t2 4 6 8 10

2

4

6

8

10

12

14

16

Inch

es o

f sn

ow

Hours

f �t� � �t,2t � 2,12t � 10,

0 ≤ t ≤ 22 < t ≤ 88 < t ≤ 9

0 < h ≤ 45h > 45

W�h� � �12h,18�h � 45� � 540,

W�45� � 570; W�50) � 660W�30� � 360; W�40) � 480;

Weight (in pounds)

Cos

t of

over

nigh

t del

iver

y(i

n do

llars

)

7.3

29.827.324.822.319.817.314.812.39.8

1 2 3 4 5 6 7 8

C

x

C � 9.8 � 2.5�x�

Weight (in pounds)

Cos

t of

over

nigh

t del

iver

y(i

n do

llars

)

48

40

32

24

16

8

2 4 6 8 10

C

x

t1 2 543

1

Time (in minutes)

Cos

t (in

dol

lars

)

2

3

4

5

C

C2

Time (in minutes)

Cos

t (in

dol

lars

)

C

t2 4 6 8 10 12

1

2

3

4

5

6

7

g�x� � �x � 1�f �x� � �x�

g�x� � x � 2f �x� � x

g�x� � 1 � �x � 2�2f �x� � x2

g�x� � 2f �x� � 2


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333202CB01a_AN.qxd 4/26/06 6:31 PM

Precalculus with Limits, Answers to Section 1.7 23C

opyr

ight

©H

ough

ton

Mif

flin

Com

pany

. All

righ

ts r

eser

ved.


Vocabulary Check (page 79)1. rigid 2. 3. nonrigid4. horizontal shrink; horizontal stretch5. vertical stretch; vertical shrink6. (a) iv (b) ii (c) iii (d) i

1. (a) (b)

(c)

2. (a) (b)

(c)

3. (a) (b)

(c)

4. (a) (b)

5. (a) (b)

(c) (d)y

x1 3 4 5

2

1

−1

−2

−3

(0, 1)

(1, 0)

(3, −1)

(4, −2)

y

x1 2 3 4 5 6

1

2

3

4

−3

(0, −2)

(1, 0)(3, 2)

(4, 4)

−2

−1

y

x1 2 3 4 5 6

1

2

3

4

−1

−2(2, −1)

(3, 0)

(5, 1)

(6, 2)

yy

x2 31 4 5

4

3

2

1

5

(0, 1)

(1, 2)

(3, 3)

(4, 4)

y

c = 1−

c = 3

c = 1

x−2−6−10

c = 3−

1086

−4−6−8

−10−12

12

c = 3 c = 1

c = 1−

c = 3−

y

c = 1−

c = 3

c = 1

x3

2

4

−4

c = 3−

4−2−3−4

y

y

x−4 −3 4

2

1

3

4c = 2

c = −2

c = 0

y

x−4 3 4

2

3

4c = 2

c = −2

c = 0y

x−4 3 4

2

3

4c = 2

c = −2

c = 0

c = −3

c = −1

c = 3

c = 1

x2 4−2

2

−2

−4

4

6

8

y

c = −3 c = −1

c = 3c = 1

x2−2−4 6 8 4

2

4

6

8

−2

−4

y

c = 3

c = 1

c = −1

c = −3x

2 8 1210

2

4

6

−2

−4

−6

y

x

c

c = 3

c = 1

c = −1

x−2−6−8

6

−2

y

y

x−2−4 42 6

6

−2

8

y

c = 3

c = 1

c = −1

y

x

c = 3

x−4 −2

−2

2 4

6

y

c = 1

c = −1

�f �x�; f ��x�

333202CB01b_AN.qxd 4/13/06 5:17 PM


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d.

(Continued)

(e) (f)

(g).

6. (a) (b)

(c) (d)

(e) (f)

(g)

7. (a) (b)

(c) (d)

(e) (f)

(g)y

x−4 −3 −2 −1 2

−3

−2

−1

1

2

3

4

5

3 4

(−1, 4)

(0, 3)

, −132( (

, 012( (

y

x−1−2 1

1

3

2

−1

−2

(−2, 2)

))(1, 0)

32

0,

)) 12

3, −

y

x2 54

1

−4

(5, 1)

(3, 0)

(2, −3)

(0, −4)

−1

−2

−3

1

y

y

x−1−2−3 2

3

4

2

−1

(−1, 3)

(−3, 4)

(0, 0)

(2, −1)

yy

x1 2−1−3

3

4

1

−1

(2, 4)

(0, 3)

(−1, 0)

(−3, −1)

y

y

x−1 41

3

4

2

−1

(−1, 4)

(1, 3)

(2, 0)

(4, −1)

1

yy

x−1 3−2

1

3

−2

−1

(−2, 3)

(0, 2)

(1, −1)

(3, −2)

y

y

x−4 −3 −2 2

−4

−3

−1

1

2

3

4

3 4

(−2, 2) (3, 2)

(−1, −2) (0, −2)

4

10

6

8

2

−2

−4

−4−6−8

−6

x4 6

(−2, 1)

(−4, −3) (6, −3)

(0, 1)

2 8

y

x

4

10

6

8

−6

42

2

−2−6−8−10

y

(−4, −1) (6, −1)

(−2, −5) (0, −5)

x4

4

10

6

8

(2, 2)

(10, −2)(0, −2)

(4, 2)

12

2

2 6

−4

−4 −2

−6

y

x−4−6−8−10

4

10

6

8

(−4, 4)

(−2, −4) (0, −4)

(6, 4)

64−2

−6

2

y

x6

4

10

6

8

(0, 2)(−2, 2)

(−4, 6) (6, 6)

2 4−2−4

−4

−6

−6−8−10

y

x

2

4

10

4 6 8 10

6

8

−4

−4−6

−6

(−6, 2)

(2, −2)(0, −2)

(4, 2)

y

y

x−1 2

−5

−4−3−2

12345

3 4 5 6 7 8 9(0, −1)

(2, 0)(6, 1)

(8, 2)

y

x−1−2−3−4−5

2

3

−2

(0, −1)

(−1, 0)(−3, 1)

(−4, 2)

y

x1 2−1−3

2

3

−1

−2

(0, 1)

(1, 2)

(−2, 0)

y

(−3, −1)

333202CB01b_AN.qxd 4/13/06 5:17 PM


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(Continued)

8.

9. (a) (b)(c) (d)

10. (a) (b)(c) (d)

11. (a) (b)

(c) (d)

12. (a) (b)(c) (d)

13. Horizontal shift of

14. Vertical shrink of

15. Reflection in the -axis of 16. Vertical shift of

17. Reflection in the -axis and vertical shift of

18. Horizontal shift of

19. (a)(b) Reflection in the x-axis, and vertical shift 12 units

upward, of (c)

(d)

20. (a)(b) Horizontal shift eight units to the right, of (c)

(d)

21. (a)(b) Vertical shift seven units upward, of (c)

(d)

22. (a)(b) Reflection in the x-axis, and a vertical shift of one

unit downward, of (c)

(d) g�x� � �f �x� � 1

1

x21−2−3

2

3

3

−2

−3

y

f �x� � x3

f �x� � x3

g�x� � f �x� � 7

x1

1−1−3−4−5−6

2345

89

1011

2 3 4 5 6

y

f �x� � x3

f �x� � x3

g�x� � f �x � 8�

8

x12 164

4

8

12

16

y

f �x� � x2

f �x� � x2

g�x� � 12 � f �x�

y

x

12

4

−48−8−12 12

−8

−12

f �x� � x2

f �x� � x2

y � �x � 2�y � �x�;

y � 1 � �xy � �x ;x

y � �x� � 4y � �x�;y � �x2y � x2 ;x

y �12 xy � x;

y � �x � 2�3y � x 3;

y � ��x � 3 � 4y � ��x � 5 � 5y � �x � 1 � 7y � �x � 3

y � ��x � 6� � 1y � �x � 2� � 4

y � ��x � 3�y � �x� � 5y � �x � 10�3 � 4y � ��x � 3�3 � 1y � �x � 1�3 � 1y � 1 � x 3

y � �x � 5�2 � 3y � ��x � 2�2 � 6y � 1 � �x � 1�2y � x2 � 1

y

x6−6

−18

−12

−6

6

12

18

(−18, −4) (18, −4)

(−9, 0) (9, 0)(0, 5)

2

x4 6 82

y

−2−4−6−8

−4

−8

−10

−12

−14

(−6, −14)

(−3, −10) (3, −10)

(6, −14)

(0, −5)(−3, 0)

6

2(3, 0)

x4 6 82

(0, 5)4

8

y

−2−4−6−8

−4

−6

−8

(−6, −4) (6, −4)

4

6

2(2, 0)

x4 6

(5, 4)

y

−4

−6

−8

−10

−2−6−10 −8

(−1, −5)

(−4, 0)

(−7, 4)

(−3, 0)

1

2

(3, 0)x

4−6−8 6 8

−1

−2

2−2−4

43( (

53

0,( (

y

−6, − 43( (6, −

4

10

6

12

(−6, 7)

(3, 3)2

8

(−3, 3)

(6, 7)

x64

y

−4−6−8−10

−4(0, −2)

x4

4

10

6

8

(2, 0)

(5, 5)

12

2

6 10

(8, 0)

8

y

−2−4

−6(−1, −4) (11, −4)

(a) (b)

(c)

(e) (f)

(d)

(g)

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d.

(Continued)

23. (a)(b) Vertical shrink of two-thirds, and vertical shift four

units upward, of (c)

(d)

24. (a)(b) Vertical stretch of two, and horizontal shift seven

units to the right, of (c)

(d)

25. (a)(b) Reflection in the x-axis, horizontal shift five units to the

left, and vertical shift two units upward, of (c)

(d)

26. (a)(b) Reflection in the x-axis, horizontal shift 10 units to the

left, and vertical shift five units upward, of(c)

(d)

27. (a)(b) Horizontal shrink of of (c)

(d)

28. (a)(b) Horizontal stretch of four, of (c)

(d)

29. (a)(b) Vertical shift two units upward, and horizontal shift one

unit to the right, of (c)

(d)

30. (a)(b) Vertical shift 10 units downward, and horizontal shift

three units to the left, of

(c)

(d)31. (a)

(b) Reflection in the x-axis, and vertical shift two unitsdownward, of f �x� � �x�

f �x� � �x�g�x� � f �x � 3� � 10

y

x−16 −12 −8

−16

−12

4

−4 4

f �x� � x3

f �x� � x3

g�x� � f �x � 1� � 2

x

1

1−1−2

2

3

4

5

2 3 4

y

f �x� � x3

f �x� � x3

g�x� � f �14x�

y

x−1 1 2 3 4

−1

1

2

3

4

f �x� � �xf �x� � �x

g�x� � f �3x�

y

x−1−2 1 2 3 4 5 6

−1

−2

1

2

3

4

5

6

f �x� � �x13,

f �x� � �x

g�x� � �f �x � 10� � 5

5

x

10

−5

−10

−10 −5−15−20

y

f �x� � x2

f �x� � x2

g�x� � 2 � f �x � 5�

x

1

1−1−2−4−5−6−7

2

3

4

−2

−3

−4

y

f �x� � x2

f �x� � x2

g�x� � 2f �x � 7�

y

x−2 2 4 6 8 10

−2

2

4

6

8

10

f �x� � x2

f �x� � x2

g�x� �23 f �x� � 4

y

x−1−2−3−4 1 2 3 4

−1

1

2

3

5

6

7

f �x� � x2

f �x� � x2

333202CB01b_AN.qxd 4/13/06 5:17 PM


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(Continued)

(c)

(d)

32. (a)(b) Reflection in the x-axis, horizontal shift five units to the

left, and vertical shift six units upward, of (c)

(d)

33. (a)(b) Reflection in the x-axis, horizontal shift four units to the

left, and vertical shift eight units upward, of (c)

(d)34. (a)

(b) Reflection in the y-axis, horizontal shift three units to theright, and vertical shift nine units upward, of

(c)

(d)35. (a)

(b) Reflection in the x-axis, and vertical shift three unitsupward, of

(c)

(d)

36. (a)(b) Horizontal shift five units to the left, and vertical

stretch of two, of (c)

(d)

37. (a)(b) Horizontal shift of nine units to the right, of (c)

(d)

38. (a)(b) Horizontal shift of four units to the left, and vertical

shift eight units upward, of (c)

(d)

39. (a)(b) Reflection in the y-axis, horizontal shift of seven units

to the right, and vertical shift two units downward, off �x� ��x

f �x� � �x

g�x� � f �x � 4� � 8

6

x42

2

6

4

12

−2−4−6

8

y

f �x� � �x

f �x� � �x

g�x� � f �x � 9�

x

15

3 6 9 12 15

3

6

9

12

y

f �x� ��xf �x� � �x

g�x� � 2 f �x � 5�

y

x−10 −8 −6 −4 −2

−6

−4

−2

2

4

2

f �x� � �x�

f �x� � �x�

g�x� � 3 � f �x�

y

x−3 −2 −1 1 2

−3

−2

1

2

3

6

3 6

f �x� � �x�

f �x� � �x�g�x� � f ��x � 3�� 9

6

x9 123

3

6

9

12

y

f �x� � �x�

f �x� � �x�g�x� � �f �x � 4� � 8

x

8

−2−4−6

−2

42

2

4

6

y

f �x� � �x�

f �x� � �x�g�x� � 6 � f �x � 5�

4

x

−2

−4

−6

−2−4−6−8−10

6

8

y

f �x� � �x�

f �x� � �x�g�x� � �f �x� � 2

x

1

1−1−2−3 32−1

−2

−3

−4

−5

y

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d.

(Continued)

(c)

(d)

40. (a)(b) Reflection in the x-axis, horizontal shift of one unit to the

left, and vertical shift six units downward, of (c)

(d)

41. (a)

(b) Horizontal stretch, and vertical shift four units down-ward, of

(c)

(d)

42. (a)(b) Horizontal shrink of and vertical shift one unit

upward, of (c)

(d)

43. 44.

45. 46.

47. 48.

49. 50.

51. (a) (b)

52. (a) (b)

53. (a) (b)

54. (a) (b)

55. Vertical stretch of

56. Vertical stretch of

57. Reflection in the -axis and vertical shrink of

58. Horizontal stretch of

59. Reflection in the -axis and vertical shrink of

60. Reflection in the -axis, vertical stretch, and vertical shifttwo units downward, of

61. 62.

63. 64.

65. (a) (b)

(c) (d)

(e) (f)y

x−6 −4 −2

−8

−6

−4

−2

2

4

6

8

2 4 6 8 10

g

y

x−2 −1 1 2

−2

−1

1

2

g

x

−2

−3

1

2

4

−2−3−4 6g

−4

−5

−6

3

4 51

y

x1−1−2−3−4−5−6

−2

−3

2 3

2

4

5

6

7

4

g

3

y

x

−2

−3

−4

−5

−6

1

2

4

−3−4 65g

3

y

x1−1−2−3−4

−2

−3

1

2 3 4 5 6

2

4

5

6

7

g

y

y � �x � 2�2 � 4y � ��x � 3

y � �x � 4� � 2y � ��x � 2�3 � 2

y � �2�x� � 2

y � �x�;x

y �12��x

y � �x ;y

y � �x�; y � �12 x�

y � �12 x2

y � x2 ;x

y � 6�x�y � �x�;y � 2x3y � x 3 ;

y � �14�xy � 8�x

y � 3�x� � 3y � �12�x�

y � �2x 3y �14 x3

y � 4x2 � 3y � �3x2

f �x� � ��x � 9f �x� � ��x � 6

f �x� � �x � 1� � 7f �x� � ��x� � 10

f �x� � ��x � 6�3 � 6f �x� � �x � 13�3

f �x� � ��x � 3�2 � 7f �x� � �x � 2�2 � 8

g�x� � f �3x� � 1

y

x1−1 2

12345678

9

3 4 5 6 7 8 9

f �x� � �x

13,

f �x� � �x

g�x� � f �12x� � 4

y

x−1

−9−8−7−6−5−4−3−2

1

1 2 3 4 5 6 7 8 9

f �x� ��x

f �x� � �x

g�x� � �f �x � 1� � 6

x4 521

1

3−1−2−3−4−5

−2

−3

−4

−5

−8

−9

y

f �x� � �x

f �x� � �x

g�x� � f�7 � x� � 2

x

4

2−2

−2

−4

2

−6

8

y

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(Continued)

66. (a) (b)

(c) (d)

(e) (f)

67. (a) Horizontal stretch of 0.035 and a vertical shift of 20.6units upward.

(b) 0.77-billion-gallon increase in fuel usage by truckseach year

(c) The graph is shifted 10 units to the left.

(d) 52.1 billion gallons. Yes.

68. (a) Horizontal shift of 20.396 units to the left and verticalshrink of 0.0054

(b) The graph is shifted 10units to the left.

69. True.

70. False. The point will lie on the transformation.

71. (a) (b)

(c)

72. If you consider the -axis to be a mirror, the graph ofis the mirror image of the graph of

73. , 74. Answers will vary.

75. 76. 77.

78. 79.

80. 81.

82.

83. (a) 38 (b) (c)

84. (a) (b) 3 (c)

85. All real numbers x except

86. All real numbers x such that except

87. All real numbers x such that

88. All real numbers x

�9 ≤ x ≤ 9

x � 8x ≥ 3,

x � 1

�x � 3�3

x2 � 12x � 38574

�4�1,x � 0,x � 1

�x � 7��x � 3�,

x � �35�x � 3�,x � 2x � 1

x�x � 2�,

�x � 4��x2 � 4x2 � 4

3x � 52�x � 5�

3x � 2x�x � 1�

�20�x � 5��x � 5�

4x�1 � x�

�0, 2��1, 1�,��2, 0�

y � f �x�.y � � f �x�x

g�t� � f �t � 2�g�t� � f �t� � 10,000g�t� �

34 f �t�

��2, �67�

��x� � �x�

f �t� � 0.0054�t � 30.396�2.

Year (0 ↔ 1990)

Am

ount

of

mor

tgag

e de

bt(i

n tr

illio

ns o

f do

llars

)

6543

7

21

42 6 8 1210t

M

f �t� � 20.6 � 0.035�t � 10�2.

Year (0 ↔ 1980)

Am

ount

of

fuel

(in

billi

ons

of g

allo

ns) 40

35

30

25

20

15

20161284t

F

y

x

g−4−8

−8

−12

−16

−20

4

8

12

16

y

x−2 −1 1 2

1

−1

2

3

4

3

g

x

3

−3

−6

−12

3 9−3−6

g

6

y

x421 3

g

7

−2

−3

6

5

4

2

3

y

−1−2−3−4−5−6

7

x421 3 5−1−2−3 6 7

6

5

4

2

1

−2

−3

g

y

x421

1

−2

−4

−5

−6

−7

−8

−9

3 5−1−2−3

g

6

y

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d.


Vocabulary Check (page 89)1. addition; subtraction; multiplication; division2. composition 3. 4. inner; outer

1. 2.

3. 4.

5. (a) (b) 4 (c)

(d) all real numbers x except

6. (a) (b) (c)


7. (a) (b) (c)


8. (a) (b) (c)(d) all real numbers x

9. (a) (b)

(c)

(d) all real numbers x such that

10. (a) (b)

(c)


11. (a) (b) (c)

(d) x; all real numbers x except

12. (a) (b) (c)


13. 3 14. 15. 5 16.17. 18. 19. 7420. 21. 26 22. 23. 24. 4325. 26.

27. 28.

29. 30.

31. (a) (b) (c)

32. (a) (b) (c)

33. (a) x (b) x (c)

34. (a) (b) (c)

35. (a) (b)Domains of and Domains of and all real numbers

36. (a) (b)Domains of and all real numbers

37. (a) (b)Domains of and all real numbersDomains of and all real numbers x such that



40. (a) (b)Domains of and all real numbersg � f :f � g,g,f,

3 � �x � 4��1 � x�g � f :f � g,g,f,

�x� � 6�x � 6�g � f :f � g,g,f,

x 4x 4

x ≥ 0f � g:gg � f :f

�x2 � 1x � 1g � f :f � g,g,f,

x � 43�x3 � 4f � g:g

x ≥ �4g � f :fx � 4�x2 � 4

x 91x3

1x3

3� 3�x � 1 � 1

9x � 20�3x20 � 3x

x4x2 � 1�x � 1�2

f �x�g�x�,f (x), g(x)

−4 14

−2

f

gf + g

10

−15 15

−10

f

g

f + g

10

−6 −4 4 6

−6

−4

−2

6

x

f

g

f + g

y

–3 –2 –1 3 4

3

−2

4

5

x

f

g

f + g

y

2 4 6

−2

2

6

8

x

f

g

f + g

y

1−2 2 3 4

1

2

3

4

x

f

g

f + g

y

35�

14�370

t 2 � 3t � 19t 2 � 3t � 5�17

�1x � 0,1

x2�x � 1�;

x 4

x � 1�x 4 � x 3 � x

x � 1x 4 � x 3 � x

x � 1

x � 0

1

x 3

x � 1

x2

x � 1

x2

�x� ≥ 2�x2 � 1��x2 � 4

x2 ;

x2�x2 � 4x2 � 1

�x2 � 4 �x2

x2 � 1�x2 � 4 �

x2

x2 � 1

x < 1�x2 � 6��1 � x

1 � x;

�x2 � 6��1 � x

x2 � 6 � �1 � xx2 � 6 � �1 � x

12 x �

54;

8x � 202x � 92x � 1

x �54

x2

4x � 5;

4x3 � 5x2x2 � 4x � 5x2 � 4x � 5

x � 22x � 52 � x

;

�2x2 � 9x � 103x � 7x � 3

x � 2x � 2

x � 2;

x2 � 42x

y

x−4 −3 −2

−4

−3

−2

1

2

3

4

1 2 3 4

h

x2−1 31

5

2

1

−2 4 65

6

7

h

y

−2 −1 1 2

−1

1

2

3

x

h

y

1 2 3 4

1

2

3

4

x

h

y

g�x�

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(Continued)

41. (a) (b)

Domains of and all real numbers x except Domains of all real numbersDomains of all real numbers x except

42. (a) (b)

Domains of and all real numbers x except Domains of all real numbersDomains of all real numbers x except

43. (a) 3 (b) 0 44. (a) (b)

45. (a) 0 (b) 4 46. (a) (b)

47.

48.

49.

50.

51.

52.

53.

54.

55.

56. (a)(b)

57. (a)

(b) is the population change in the year 2005.

58. (a)(b) Number of dogs and cats in the year 2005.(c) The function represents the number of dogs and

cats per capita.

59. (a)

(b)

60. (a) which represents the

millions of dollars spent on exercise equipment com-pared with the millions of people in the United States.

(b)61. (a)

(b) this amountrepresents the amount spent on health care in theUnited States.

(c)

(d) In 2008, $1298.708 billion is estimated to be spent on health services and supplies, and in 2010, $1505.4billion is estimated.

62. (a) For each time there corresponds one and only onetemperature

(b)(c) All the temperature changes occur 1 hour later.(d) The temperature is decreased by 1 degree.

(e)

63. (a) (b)

(c) represents the area of the

circular base of the tank on the square foundation withside length

64. represents the area of thecircle at time

65. (a) This represents the num-ber of bacteria in the food as a function of time.

(b)66. (a) represents the cost

of t hours of production.

(b) t � 4.75 hours

�C � x��t��C � x��t� � 3000t � 750;t � 2.846 hours

N�T�t�� 30�3t2 � 2t � 20�t.

�A � r��t��A � r��t� � 0.36� t2;x.

�A � r��x��A � r��x� � �x

22;

A�r� � �r 2r (x) �x

2

T�t� � �60,

12t � 12,

72,

�12t � 312,

60,

0 ≤ t ≤ 6

6 < t < 7

7 ≤ t < 20

20 < t < 21

21 ≤ t ≤ 24

60�, 72�

T.t

0100

720

y1

y3

y2

y1 + y2 + y3

y1 � y2 � y3 � 2.892t2 � 6.55t � 479.6;y3 � �0.465t2 � 9.71t � 7.4y2 � 3.357t2 � 26.46t � 379.5y1 � 10.20t � 92.7

h�12� � 14.98h�10� � 12.90;h�7� � 11.02;

h�t� �25.95t2 � 231.2t � 3356

3.02t � 252.0,

�A � N��12� � 123.84�A � N��8� � 81.44�A � N��4� � 84.16�A � N��t� � 1.41t2 � 17.6t � 132�A � N��12� � 878.64�A � N��8� � 861.84�A � N��4� � 1014.96�A � N��t� � 5.31t2 � 102.0t � 1338

h�t�p�5� �

p�t� � d�t� � c�t�c�5�

c�t� �p�t� � b�t� � d�t�

p�t� � 100

060

800

R2

R1

R3

R3 � 734 � 7.22t � 0.8t2

10 20 30 40 50 60

50

100

150

200

250

300

x

Speed (in miles per hour)

Dis

tanc

e tr

avel

ed(i

n fe

et)

R

B

T

T �34 x �

115 x2

g�x� � x 3f �x� �27x � 6 3�x10 � 27x

,

g�x� � �x2f �x� �x � 34 � x

,

g�x� � 5x � 2f �x� �4x2,

f (x) �1

x, g(x) � x � 2

g�x� � 9 � xf �x� � �x,

f (x) � 3�x, g(x) � x2 � 4

g�x� � 1 � xf �x� � x 3,

f (x) � x2, g(x) � 2x � 1

22

0�1

x � 0, �2f � g :g:

x � ±1g � f :f

x2 � 2x2 � 1

3x2 � 2x

x � �3f � g:g:

x � 0g � f :f

1x

� 31

x � 3

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d.

(Continued)

67. represents 3 percent of an amount over $500,000.

68. (a) (b)

(c) represents thedealership discount after the factory discount.

represents thedealership discount after the factory discount.

(d)

$16,450 is the lower cost because 10% of the price ofthe car is larger than $2000.

69. False. and

70. True. The range of must be a subset of the domain of for to be defined.

71. Answers will vary. 72. Proofs will vary.

73. 3 74. 75.

76.

77. 78.

79. 80.

2

x102−2

−2

−4

−8

4

84

(7, 0)

y

x2−2 4 6

2

−210 12

4

6

8

10

(8, −1)

y

5x � 7y � 35 � 03x � 2y � 22 � 0

x

4

−3

−3

−4

−1 1−4−5−6−7

3

2

1

(−6, 3)

y

x42−2−4 6 8

2

−4

−6

−2

−8

−10

10

(2, −4)

y

x � y � 3 � 03x � y � 10 � 0

�2�x � h� � 1 � �2x � 1h

�2

�2�x � h� � 1 � �2x � 1

�4x�x � h��2x � h

� f � g��x�fg

�g � f ��x� � 6x � 6� f � g��x� � 6x � 1

�S � R��p� � $16,650�R � S��p� � $16,450

�S � R��p��S � R��p� � 0.9�p � 2000�;

�R � S��p��R � S��p� � 0.9p � 2000;

S�p� � 0.9pR�p� � p � 2000

g� f �x��

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Vocabulary Check (page 99)1. inverse; f-inverse 2. range; domain3. 4. one-to-one 5. horizontal

1. 2. 3.

4. 5.

6. 7. 8.

9. c 10. b 11. a

12. d

13. (a)

(b)

14. (a)

(b)

15. (a)

(b)

16. (a)

(b)

17. (a)

(b)

18. (a)

(b)

19. (a)

(b)

2 4 6 8 10

10

2

4

6

8

x

f

g

y

� ��x � 4 �2� 4 � x

g� f �x�� g��x � 4� � ��x2 � 4� � 4 � x

f �g�x�� f �x2 � 4�, x ≥ 0

1 2 3

1

2

3

x

f = g

y

g� f �x�� g1

x �1

1�x� x

f �g�x�� f 1

x �1

1�x� x

y

x

g

f

−3−4 1 2 3 4−1

−3

−4

1

2

3

4

g� f �x�� gx3

8 � 3�8x3

8 � x

f �g�x�� f � 3�8x � �� 3�8x �3

8� x

−8 −6 −4 −2 2 4

−8

−6

−4

2

4

x

f

g

y

g� f �x�� g�3 � 4x� �3 � �3 � 4x�

4� x

f �g�x�� f 3 � x

4 � 3 � 43 � x

4 � x

y

x

g

f

1 2 3 4 5

1

2

3

4

5

g� f �x�� g�7x � 1� ��7x � 1� � 1

7� x

f �g�x�� fx � 17 � 7x � 1

7 � 1 � x

−8 −4 −2 2 6 8

−8

−4

2

6

8

x

g

f

y

g� f �x�� g�x � 5� � �x � 5� � 5 � x f �g �x�� f �x � 5� � �x � 5� � 5 � x

–3 –2 1 2 3

–3

–2

1

2

3

x

f

g

y

g� f �x�� g�2x� ��2x�

2� x

f �g�x�� fx2 � 2x

2 � x

f �1�x� � 5�xf �1�x� � x 3f �1�x� � 5x � 1

f �1�x� �x � 1

3f �1(x) � x � 4

f �1�x� � x � 9f �1�x� � 3xf �1�x� �16 x

y � x

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d.

(Continued)

20. (a)

(b)

21. (a)

(b)

22. (a)

(b)

23. (a)

(b)

24. (a)

(b)

25. No 26. Yes27.

28.

29. Yes 30. No 31. No 32. Yes33. 34.

The function has The function does not havean inverse. an inverse.

−12 12

−2

14

−4 8

−4

4

x4−2−4

−4

−6

2

6 8

4

6

ffg

g

y

�2x � 6 � 3x � 6x � 3 � x � 2

� x

g� f �x�� gx � 3x � 2 �

2x � 3x � 2 � 3

x � 3x � 2 � 1

�2x � 3 � 3x � 32x � 3 � 2x � 2

� x

f �g�x�� f 2x � 3x � 1 �

2x � 3x � 1 � 3

2x � 3x � 1 � 2

x

6

2

4

2−6−8−10 4 6 8 10

g−6

−10

8

g

f

f

10

−4

−8

y

��5x � 5 � x � 5

x � 1 � x � 5� x

g� f �x�� gx � 1x � 5 �

�5x � 1x � 5 � 1

x � 1x � 5

� 1

��5x � 1 � x � 1

�5x � 1 � 5x � 5� x

f �g�x�� f �5x � 1x � 1 �

�5x � 1x � 1 � 1

�5x � 1x � 1 � 5

1 2 3 4 5

1

2

3

4

5

x

f

g

y

�1 � 1��1 � x�

1��1 � x��

1 � x � 1

1� x

x ≥ 0 g� f �x�� g 1

1 � x,

�1

1 � �1 � x��x�

x

x � 1 � x� x

0 < x ≤ 1 f �g�x�� f 1 � x

x ,

−12 –9 –6 –3 6 9 12

–12

–9

–6

6

9

12

x

f

g

y

� �9 � �9 � x2� � x

g� f �x�� g�9 � x2�, x ≥ 0

� 9 � ��9 � x �2� x

f �g�x�� f ��9 � x �, x ≤ 9

−6 −4 −2 6

−6

−4

−2

6

x

f

g

y

� 3�1 � �1 � x3� � x g � f �x�� g�1 � x3�

� 1 � � 3�1 � x �3� x

f �g �x�� f � 3�1 � x �

10 7 4 1 2 5

3 2 1 0 1 2��f �1�x�

��x

0 2 4 6 8

0 1 2 3�1�2f �1�x�

�2x

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(Continued)

35. 36.

The function does not The function has anhave an inverse. inverse.

37. 38.

The function does not The function does nothave an inverse. have an inverse.

39. (a)

(b)

(c) The graph of is the reflection of the graph of fin the line

(d) The domains and ranges of f and are all real numbers.

40. (a)

(b)

(c) The graph of is the reflection of the graph of f inthe line


41. (a)(b)



42. (a)(b)



43. (a)(b)


(d) The domains and ranges of f and are all real num-bers x such that

44. (a)(b)

1 2 3 4 5

1

2

3

4

5

x

f

y

f −1

f �1�x� � �x

x ≥ 0.f �1

y � x.f �1

1 2 3 4 5

1

2

3

4

5

x

f −1

f

y

f �1�x� � x2, x ≥ 0

f �1

y � x.f �1

−6 −4 2 4 6

−6

2

4

6

x

f −1

f

y

f �1�x� � 3�x � 1

f �1

y � x.f �1

x

3

2

2−1−3 3

f

−1

−3

f −1

y

f �1�x� � 5�x � 2

f �1

y � x.f �1

x

3

2

2

f

−2

−2 −1−3

−3

f −11

1

y

3

f �1�x� �x � 1

3

f �1

y � x.f �1

–2 2 4 6 8

2

−2

4

6

8

x

f −1

f

y

f �1�x� �x � 3

2

−24 24

−8

24

−12

−20

12

20

−10 2

−4

4

−10 10

−10

10

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d.

(Continued)


(d) The domains and ranges of f and are all real numbers such that

45. (a)(b)

(c) The graph of is the same as the graph of f.(d) The domains and ranges of f and are all real num-

bers x such that

46. (a)(b)


(d) The domain of f and the range of are all real num-bers x such that The domain of and therange of f are all real numbers x such that

47. (a)

(b)

(c) The graph of is the same as the graph of f.(d) The domains and ranges of f and are all real num-

bers x except

48. (a)

(b)

(c) The graph of is the same as the graph of f.

(d) The domains and ranges of f and are all real num-bers x except

49. (a)

(b)


(d) The domain of f and the range of are all real num-bers x except The domain of and the rangeof f are all real numbers x except

50. (a)

(b)


(d) The domain of f and the range of are all real num-bers x except The domain of and therange of f are all real numbers x except x � 1.

f �1x � �2.f �1

y � x.f �1

4

6

x2 4 6

f

f

y

f −1

f −1

f �1�x� ��2x � 3

x � 1

x � 1.f �1x � 2.

f �1

y � x.f �1

x

6

4

4 6

f

−2

−6

f −1

f −1f

2

−4

−2−4−6

y

f �1�x� �2x � 1x � 1

x � 0.f �1

f �1

x1

1

−1

−2

−3

−3 −2 −1 2

2

3

y

f = f −1

f �1�x� � �2x

x � 0.f �1

f �1

–3 –2 –1 1 2 3 4

–3

–2

1

2

3

4

x

f = f −1

y

f �1�x� �4x

x ≥ �2.f �1x ≤ 0.

f �1

y � x.f �1

x

4

2

2

f

−4

−3

−3−4 3

1

1 4

3

y

f −1

f �1�x� � ��x � 2

0 ≤ x ≤ 2.f �1

f �1

1 2 3

1

2

3

x

f = f −1

y

f �1�x� � �4 � x2, 0 ≤ x ≤ 2

x ≥ 0.f �1

y � x.f �1

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(Continued)

51. (a)

(b)



52. (a)

(b)



53. (a)

(b)


(d) The domain of f and the range of are all real num-bers x except The domain of and the rangeof f are all real numbers x except

54. (a)

(b)


(d) The domain of f and the range of are all real num-bers x except The domain of and therange of f are all real numbers x except

55. No inverse 56. No inverse 57.

58. 59. No inverse

60. 61.

62. No inverse 63. No inverse 64. No inverse

65. No inverse 66.

67.

68. 69. 32 70. 0

71. 600 72. 73. 74.

75. 76. 77. 78.

79. (a)(b) represents the year for a given number of house-

holds in the United States.(c)

(d) (e)

(f) the results are similar.

80. (a) Yes(b) represents the time in years for a given sales level.(c)(d) No, it wouldn’t exist because

81. (a) Yes(b) yields the year for a given number of miles trav-

eled by motor vehicles.(c)(d) No. would not pass the Horizontal Line Test.f (t�

f �1�2632� � 8

f �1

f �12� � 2794. f�1�1825� � 10f �1

f�1�108,209� � 11.418;

f �1�117,022� � 17f�1 �t � 90,183.63

1578.68

y � 1578.68t � 90,183.63

f �1

f �1�108,209� � 11

x � 3

2x � 1

2x � 3

2x � 1

2

2 3�x � 32 3�x � 39��4

x ≥ 0f �1�x� � x 2 � 2,

x ≥ 0f �1�x� �x2 � 3

2,

x ≥ 0f �1�x� � 2 � x,

f �1�x� � �x � 3f �1�x� �5x � 4

3

f �1�x� �x � 5

3

g�1�x� � 8x

x � �4.f �1x � �3.

f �1

y � x.f �1

4

x8 12

f

f

16

8

y

f −1f −1

f �1�x� ��6x � 4

2x � 8

x �32.f �1x � �

54.

f �1

y � x.f �1

x

3

2

2

f

−3

3

f −1

f −1

−3

f1

−2

−2 1

y

f �1�x� �5x � 46 � 4x

f �1

y � x.f �1

1 2 3

1

2

−2

−2−3

−3

3

x

f

y

f −1

f �1�x� � x5�3

f �1

y � x.f �1

−4−6 2 4 6

−6

2

4

6

x

f −1

f

y

f �1�x� � x 3 � 1

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d.

(Continued)

82. (a)

(b) number of units produced; hourly wage(c) 19 units

83. (a)

degrees Fahrenheit; % load(b) (c)

84. (a)

number of pounds of the lessexpensive ground beef

(b) (c) 20 pounds

85. False. has no inverse.

86. True. If and then the -intercept of is and the -intercept of is

87–88. Answers will vary.

89.

90.

91.

92.

The graph of does not pass the Horizontal Line Test, sodoes not exist.

93. 94. 95. 96.

97. 98. 99. 100.

101. 102. 103. 16, 18

104. b � �10 feet, h � 2�10 feet

1185, �10

3

�1, 33 ±�5�1, �13

32

5 ± 2�2±8k � �12k �

14

f �1�x�f

–3 –2 –1 1 2 3 4 5 6

–3

–2

2

3

4

5

6

x

y

1

x321

3

5

2

4

−3−1

−2

−3

−4−5

y

4

x6 82

2

4

6

8

y

��6, 0�.f �1x�0, �6�fy

f �1�x� � x � 6,f �x� � x � 6

f �x� � x2

62.5 ≤ x ≤ 80

y �x � total cost;

y �80 � x

0.35

0 < x < 92.11

0 6000

100

y �x �

y ��x � 245.500.03

, 245.5 < x < 545.5

x �y �

y �x � 80.75

1 3 4 6

1 2 6 7y

x 1 2 6 7

1 3 4 6f �1�x�

x

2 3

1 3�1�2f �1�x�

�2�5x

1 3

2 3�2�5y

�1�2x

2 1 3 4

6 0 2 3��y

��x

3 2 0 6

4 3 1 2��f �1�x�

��x

0 3

3 4 0 �1y

�2�4x

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Vocabulary Check (page 109)1. variation; regression 2. sum of square differences3. correlation coefficient 4. directly proportional5. constant of variation 6. directly proportional7. inverse 8. combined 9. jointly proportional

1.

The model is a good fit for the actual data.

2.

The model is not a good fit. Answers will vary.

3. 4.

5. 6.

7. (a) and (b)

(c)

(d) The models are similar.

(e) Part (b): 238 feet; Part (c): 241.51 feet

(f) Answers will vary.

8. (a) and (b)

(c)

(d) The models are similar.

(e) Part (b): $3163.6 million; Part (c): $3164.6 million

(f) Answers will vary.

9. (a) (b)

(c)

The model is a good fit.

(d) 2005: $800 million; 2007: $876.8 million

(e) Each year the annual gross ticket sales forBroadway shows in New York City increase by$38.4 million.

10. (a) y � 0.43x � 67.7

50

14

800

S � 38.4t � 224

50

14

800

y � 251.56t � 608.8

y � 251.5x � 608.9

y

t

Tota

l rev

enue

s(i

n m

illio

ns o

f do

llars

)

Year (5 ↔ 1995)

5 7 9 11 13

500

1000

1500

2000

2500

3000

y � 1.03t � 130.27

y

t

Len

gth

(in

feet

)

Year (12 ↔ 1912)

12 36 60 84 108

140

160

180

200

220

240

y �13x � 2y � �

12x � 3

1 2 3 4 5

1

3

4

5

x

y

1 2 3 4 5

1

2

4

5

x

y

y � �54x � 8y �

14x � 3

1 2 3 4 5

1

2

3

4

5

x

y

1 2 3 4 5

1

2

4

5

x

y

y

tWin

ning

tim

e (i

n m

inut

es)

Year (0 ↔ 1950)

0 8 16 24 32 40 48 56

3.84.04.24.44.64.85.0

5.4

y

t2 4 6 8 10 12

125,000

130,000

135,000

140,000

145,000

Num

ber

of e

mpl

oyee

s(i

n th

ousa

nds)

Year (2 ↔ 1992)

333202CB01b_AN.qxd 4/13/06 5:17 PM


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d.

(Continued)

(b) (c) 106.4 million

(d) For every million households with cable TV, there is a0.43 million increase in the number of households withcolor TV.

11. Inversely 12. Directly

13.

14.

15.

16.

17.

18.

19.

20.

xx2 4 6 8 10

1

3

5

2

2

2

2

1

y

xx2 4 6 8 10

1

2

3

5

4

4

4

4

1

y

xx2 4 6 8 10

10

10

10

10

10

1

2

3

4

5

y

2 4 6 8 10

5

10

15

20

25

x

y

2 4 6 8 10

10

20

30

40

50

x

y

2 4 6 8 10

40

80

120

160

200

x

y

2 4 6 8 10

20

40

60

80

100

x

y

6090

90

110

x 2 4 6 8 10

120

564

536

516

54y � k�x2

x 2 4 6 8 10

150

132

118

18

12y � k x2

x 2 4 6 8 10

1 4 9 16 25y � kx2

x 2 4 6 8 10

110

532

518

58

52y � k x2

x 2 4 6 8 10

5 15

516

59

54y � k�x2

x 2 4 6 8 10

4 16 36 64 100y � kx2

bx 2 4 6 8 10

8 32 72 128 200y � kx2

x 2 4 6 8 10

2 8 18 32 50y � kx2

333202CB01b_AN.qxd 4/13/06 5:17 PM


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(Continued)

21. 22. 23.

24. 25. 26.

27. 28. 29.

30.

31. Model: 25.4 centimeters, 50.8 centimeters

32. Model: ; 18.9 liters, 94.6 liters

33. 34. $37.84

35. (a) 0.05 meter (b) newtons

36. newtons 37. 39.47 pounds

38. Combined lifting force pounds

39. 40. 41.

42. 43. 44.

45. 46. 47.

48.

49. The area of a triangle is jointly proportional to its base andheight.

50. The surface area of a sphere varies directly as the square ofits radius.

51. The volume of a sphere varies directly as the cube of itsradius.

52. The volume of a right circular cylinder is jointly propor-tional to the product of its height and the square of itsradius.

53. Average speed is directly proportional to the distance andinversely proportional to the time.

54. varies directly as the square root of and inversely as thesquare root of

55. 56. 57.

58. 59. 60.

61. 62. 63. mile per hour

64. 65. 506 feet

66. Diameter 67. 1470 joules68. No. The 15-inch pizza is the best buy.69. The velocity is increased by one-third.70. (a) The safe load is unchanged.

(b) The safe load is eight times as great.(c) The safe load is four times as great.(d) The safe load is one-fourth as great.

71. (a)

(b) Yes.

(c)

(d) (e)

72. (a) (b) Yes. (c) 15.7 pounds

73. (a)

(b) 0.2857 microwatt per square centimeter

74. The illumination is reduced to one-fourth of its originalvalue. Explanations will vary.

75. False. y will increase if k is positive and y will decrease ifk is negative.

76. False. E is jointly proportional to the mass of an object andthe square of its velocity.

77. True. The closer the value of is to 1, the better the fit.�r�

250

55

0.2

k � 0.575

F

1

2

3

4

5

7

6

Force (in pounds)

y

Len

gth

(in

cent

imet

ers)

2 4 6 8 10 12

1433 meters

00

6000

6

C �4300

d

k4 � 4800, k5 � 4500k1 � 4200, k2 � 3800, k3 � 4200,

Tem

pera

ture

(in

°C)

d

C

2000 4000

1

2

3

4

5

Depth (in meters)

� 2r � 0.054 inch

k�2v�2

kv2 � 4

0.61v �24pq287s2z �

2x2

3y

P �18xy2F � 14rs 3z � 2xy

y �28

xy �

75x

A � �r2

W.g

R � kS�L � S�

F �km1m2

r2R � k�T � Te�P �

k

V

z � kx 2y3F �kg

r2h �

k�s

y �k

x2V � ke3A � kr2

� 2F � 120

29313

17623

y � 0.07x;y � 0.0368x; $7360

y �5314 x

y �3313 x;

I � 0.0375P

I � 0.035Py �290

3 xy � 205x

y � 7xy �12

5xy �

120

x

y � �7

10xy �

2

5xy �

5

x

2 4 6 8 10

1

2

3

4

5

x

y

333202CB01b_AN.qxd 4/13/06 5:17 PM


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(Continued)

78. (a) Good approximation (b) Poor approximation(c) Poor approximation (d) Good approximation

79. The accuracy is questionable when based on such limiteddata.

80. Answers will vary.81. 82.

83. 84.

85. (a) (b) (c) 21

86. (a) 6 (b) 9 (c) 383 87. Answers will vary.

�73�

53

1

x

2 3 40−1−2

13

−−4 −3 −2 −1 0 1 2 3 4 5

x

x ≤ �13x ≥ 3,�4 < x < 5

1

x

2 3 540−1

1183 4 5 6 7 8 9

x

x ≥ 118x > 5

333202CB01b_AN.qxd 4/13/06 5:17 PM

Precalculus with Limits, Answers to Review Exercises 43C

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Review Exercises (page 117)1. 2.

3. Quadrant IV

4. Quadrant I or II

5. (a) (b) 5(c)

6. (a) (b)

(c)

7. (a) (b) 9.9(c)

8. (a)

(b)(c)

9.

10.

11. $656.45 million

12. (a)

(b)

13.14. (a)

(b)

15.

16.

17.

x−2

−2−4−6 2 4

4

6

8

10

y

–3 –2 –1 1 2 3

–5

–4

–3

–2

–1

1

x

y

l � 24 inches, w � 8 inches, h � 12 inches

h

w

l

Radius 22.5 centimeters

80�F

Actual temperature (in °F)

App

aren

t tem

pera

ture

(in

°F)

8070

90

110

130

150

100

120

140

7065 75 85 9580 90 100

y

x

��4, 6�, ��1, 8�, ��4, 10�, ��7, 8�

�2, 5�, �4, 5�, �2, 0�, �4, 0��1.8, �0.6��14.4

x

1

1

(0, −1.2)

(−3.6, 0)

−1−4

−2

−3

−4

y

�2.8, 4.1�

x2 4 6

2

4

8

6

−2

(0, 8.2)

(5.6, 0)

y

�1, 32�3�13

x62

2

−2

−2−4

−4

6

(4, −3)

(−2, 6)

4

y

��1, 132 �

x42

2

4

8

−2−4

(1, 5)

(−3, 8)

y

−2−4 2 4 6 8−2

−4

−6

2

4

6

y

x

y

x−2−4−6 2 4 6 8

−2

−4

−6

−8

2

4

6

0 1 2

1�2�5�8�11y

�1�2x

0 2 4

4 3 2 1 0y

�2�4x

0 1 2 3 4

4 0 0 4�2�2y

�1x

333202CB01b_AN.qxd 4/13/06 5:17 PM

Precalculus with Limits, Answers to Review Exercises 44

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d.

(Continued)

18.

19. 20.

21. 22.

23. 24.

25. intercept: 26. intercepts:intercept: intercept:

27. x-intercepts: y-intercept:

28. intercepts:intercept:

29. No symmetry 30. No symmetry

31. y-axis symmetry 32. y-axis symmetry

33. No symmetry 34. No symmetry

35. No symmetry 36. y-axis symmetryy

x−3−6−9 3 6 9

−3

3

6

9

12

15

x1

7

−6−1

−−5

1

3

4

5

6

−1−24 2−3

y

x

2

42 6−4−6

−8

−10

y

x1 2 4

7

−13−2−3−4

1

2

4

5

6

y

x

2

42−4−6 6 8

−6

−12

−4

−2

y

x1

6

−13 42−1−3−4

3

−2

1

2

4

y

x6

1

2 3 4 5−1−2−1

−2

−3

−4

−5

−6

−7

y

x1−1−2−3−4

4

2−1

3

1

4

−2

−3

−4

y

�0, 0�y-�2, 0�, ��2, 0�, �0, 0�x-

�0, 5��1, 0�, �5, 0�

�0, �2�y-�0, 7�y-�2, 0�, ��4, 0�x-��7

2, 0�x-

–2 –1 1 2 3 5 6

–4

–3

–2

x

y

–3 –2 –1 1 2 3

–5

–4

–3

–2

1

x

y

−3 −2 −1 1 2 3 4 5

2

−2

3

4

5

6

x

y

–2 –1 1 2 3 4 5 6

–2

1

3

4

5

6

x

y

–5 –4 –3 1 2 3

–6

–5

–4

–3

1

2

x

y

–5 –4 –3 –1 1 2 3

–2

1

3

4

5

6

x

y

x

−3

5431

1

−2

−4

−9

−3−4−5 −1

y

–3 –2 –1 1 2 54

–3

–2

4

5

x

y

0 1 2 3

1 6�3�8�9�6y

�1�2x

333202CB01b_AN.qxd 4/13/06 5:17 PM


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(Continued)

37. Center: 38. Center: Radius: 3 Radius: 2



43.

44.

45. (a)

(b)

(c) 12.5 pounds

46. (a)

(b) 1999

47. slope: 0 48. slope: undefinedy-intercept: 6 y-intercept: none

49. slope: 3 50. slope: y-intercept: 13 y-intercept: 9

51. 52.

53. 54.

m � 0m � �5

11

−4 −2 2 4 6 8

–4

–2

4

6

8

x

(−3, 2) (8, 2)

y

x

(−4.5, 6)

(2.1, 3)

y

−2−2

2

6

8

−4

2 4 6−4−6

m � �37m � �

12

x

6

42

4

6

2

−2−2

−4−6

(−1, 8)

(6, 5)

10

y

x4

4

(−7, 1)

(3, −4)

2−6−8

2

−4

−6

−8

y

x

12

9

63

6

9

3

−3−3

−6−9

y

x

12

−9 −3−3

63−6

3

6

9

−6

y

�10

y

x

4

2

2 3

1

1 4

3

−1−1−2−4

−2

−3

−4

x

y

−2

−2

2

4

8

2 4 6−4

t

1800

Num

ber

of T

arge

t sto

res

Year (4 ↔ 1994)

4 6 8 10 12

800

1000

1200

1400

1600

N

x

F

4 8 12 2416 20

5

10

15

20

25

Length (in inches)

Forc

e (i

n po

unds

)

30

�x � 1�2 � � y �132 �2

�854

�x � 2�2 � �y � 3�2 � 13

x9

3

3−3−6−9−3

−15

6

9

15

( (32

−4,

−6

y

x2

8

8−2

−8

2

4

−4

−2−4−6 4

( (12, −1

y

��4, 32�;�12, �1�;

x4

8

10

4

−2−4−6−8

2

6

(0, 8)

6 8

10

12

14

16

18

y

–8 –4 –2 4

–6

–2

2

6

x(−2, 0)

y

�0, 8�;��2, 0�;

x

3

3−3

1

1−1−1

−3

(0, 0)

y

–4 –2 –1 1 2 4

–4

–2

–1

1

2

4

x(0, 0)

y

�0, 0�;�0, 0�;

x 0 4 8 12 16 20

F 0 5 10 15 20 25

333202CB01b_AN.qxd 4/13/06 5:17 PM


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d.

(Continued)

55. 56.

57. 58.

59. 60.

61. 62.

63. (a) (b)

64. (a) (b)

65.

66.

67. No 68. Yes 69. Yes 70. No

71. (a) 5 (b) 17 (c) (d)

72. (a) (b) (c) 2 (d) 6

73. All real numbers x such that

74. All real numbers

75. All real numbers x except

76. All real numbers

77. (a) 16 feet per second (b) 1.5 seconds(c) feet per second

78. (a)(b) Domain: Range: (c) liters

79.

80.

81. Function 82. Function 83. Not a function

84. Not a function 85. 86.

87. 88. 1,

89. Increasing on Decreasing on Constant on

90. Increasing on and Decreasing on and

91. 92.

93. 94.

95. 4 96. 28 97.

98. 99. Neither even nor odd100. Even 101. Odd 102. Even

�0.2

1 � �22

−9

9−6

1

(2.54, −7.88)

(0.13, −0.94)

0 75

0.75−0.75

0.25(0.1250, 0.000488)

−7

6−6

1

(−1.41, −6) (1.41, −6)

(0, −2)

−1

3−3

3

(1, 2)

�0, 2��, �2��2, ��2, 0�

��1, 0��, �1�

�0, ��±5�

38

�1, 1573, 3

3x2 � 3xh � h2 � 10x � 5h � 1, h � 0

4x � 2h � 3, h � 0

813

20 ≤ y ≤ 500 ≤ x ≤ 50;f �x� � 0.4�50 � x� � x � 20 � 0.6x

�16

t

4

421

1

−2

−1−2−3−4−5

5

3

2

3

6

7

y

x4

6

−6

2

6

−4

−2

4

y

x � 3, �2

x

6

4−2

−4−6 2

4

6

8

10

y

x

y

−6 −4 −2−2

2

6

8

10

2 4 6

�5 ≤ x ≤ 5

�1�3

t2 � 2t � 2t 4 � 1

6 ≤ t ≤ 11V � 5.15t � 42.05,

6 ≤ t ≤ 11V � 850t � 7400,

y �32x � 15y � �

23x �

73

y � �45x �

25y �

54x �

234

y � �15x �

15y � �

43x �

83

y �32x � 2x � 0

x

8

4

2

2

4

6

−2−2

−4

−6

−8

−4−6−10−12

(−8, 5)

y

x

(10, −3)

−8

−6

−4

−2−2 2 4 8 10 12

4

6

y

x � �8y � �12x � 2

–2

2

4

10

x

(−2, 6)

y

−6 −4 −2 2 4 6

x

(0, −5)

y

−4 −2

−2

2

−4

−8

2 4 6

y � 6y �32x � 5

333202CB01b_AN.qxd 4/13/06 5:17 PM


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(Continued)

103. 104.

105. 106.

107. 108.

109. 110.

111. 112.

113. 114.

115. 116.

117. (a)(b) Vertical shift of nine units downward(c)

(d)

118. (a)(b) Horizontal shift of two units to the right and vertical

shift of two units upward(c)

(d)

119. (a)(b) Horizontal shift of seven units to the right(c)

(d)120. (a)

(b) Horizontal shift of three units to the left and verticalshift of five units downward

f �x� � �x�h�x� � f �x � 7�

x2 4 6 8 10 12

2

6

4

10

8

12

h

y

f �x� � �x

h�x� � f �x � 2� � 2

x

1

1−1

2

3

4

−2

5

2 3 4−1

h

y

f �x� � x3

h�x� � f �x� � 9

x

2

4

−10

2 4 6−6 −42

h

y

f �x� � x2

y � �xy � x3

x4−2

−4

−6

−8

−4−6−8

2

2

4

6 8

6

8

y

x9

6

3

−15

3

6

−12

−3 12 15−6−9−12

y

y

x−5 −4 −3 −2

−2

−1

1

2

4

−1 1

y

x−3 −2 −1

−6

−5

−2

1

2

3

3 4 5 6

y

x−10 −8 −4

−6

−4

−2

2

4

6

−2 2

y

x1−1 2 3 4 5 6

123456

y

x−2

−1

2

3

−1 1 2

y

x−3 −2 −1

−6

−5

−4

−3

−2

1

2

3

2 3 4 5 6

y

x−2

−3

−1

1

−1 1 2

y

x−4−6

−2

−4

−6

4

6

4 6

y

x−1−2−3−4−5−7 1

−2

−3

−4

−5

−6

1

2

y

x−4

−4

−3

−2

−1

3

4

−3 −2 −1 1 2 3 4

f �x� � �34 x � 5f �x� � �3x

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d.

(Continued)

(c)

(d)

121. (a)(b) Reflection in the x-axis, horizontal shift of three units

to the left, and vertical shift of one unit upward(c)

(d)

122. (a)(b) Reflection in the x-axis, horizontal shift of five units

to the right, and vertical shift of five units downward(c)

(d)

123. (a)(b) Reflection in the x-axis and vertical shift of six units

upward(c)

(d)

124. (a)

(b) Reflection in the x-axis, horizontal shift of one unit tothe left, and vertical shift of nine units upward

(c)

(d)

125. (a)(b) Reflections in the x-axis and the y-axis, horizontal

shift of four units to the right, and vertical shift of sixunits upward

(c)

(d)

126. (a)(b) Reflection in the x-axis, horizontal shift of one unit to

the left, and vertical shift of three units downward(c)

(d)

127. (a)(b) Horizontal shift of nine units to the right and vertical

stretch(c)

(d) h�x� � 5 f �x � 9�

y

x−2 2 4 6 10 12 14

−5

−10

−15

5

10

15

20

25

h

f �x� � �x�

h�x� � �f �x � 1� � 3

y

x−6

−10

−8

−6

−4

−2

2

−4 −2 2 4 6

h

f �x� � x2

h�x� � �f ��x � 4� � 6

x

2

2

4

6

8

−4

10

4 86−2

h

y

f �x� � �x�h�x� � �f �x � 1� � 9

x6

2

−2−4 42

4

10

6 h

y

f �x� � �x

h�x� � �f �x� � 6

y

x−2 −1−3 1 2

−3−2

123456

3 4 5 6 9

9

h

f �x� � �x�h�x� � �f �x � 5� � 5

2

x102

−2−2

−4

−6

−8

4

84 6

h

y

f �x� � x3

h�x� � �f �x � 3� � 1

x4

4

−8

−6−8 2

2

−6h

y

f �x� � x2

h�x� � f �x � 3� � 5

x

2

4

−2−4−8−10

6

2 4−2

−4

−6

−8

h

y

333202CB01b_AN.qxd 4/13/06 5:17 PM


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(Continued)

128. (a)(b) Reflection in the x-axis and vertical shrink(c)

(d)129. (a)

(b) Reflection in the x-axis, vertical stretch, and horizontalshift of four units to the right

(c)

(d)130. (a)

(b) Vertical shrink and vertical shift of one unit downward(c)

(d)

131. (a) (b)

(c)


132. (a) (b)

(c)


133. (a) (b)

Domains of and all real numbers134. (a) (b)

Domains of all real numbers

135.136.137. (a)

(b)

(c)138. (a)

The composition function represents the num-ber of bacteria in the food as a function of time.

(b)

139. 140.

141. The function has an inverse.

142. The function does not have an inverse.

143. The function has an inverse.

144. The function does not have aninverse.

145. 146.

The function has an The function has aninverse. inverse.

147. (a)(b)

(c) The graph of is the reflection of the graph of inthe line

(d) Both f and have domains and ranges that are allreal numbers.

f �1

y � x.ff �1

x

8

2 f

−6

−10

−2−6−10 −8

f −1

8

6

−8

y

f �1�x� � 2x � 6

−2

4−8

6

−4

8−4

4

−2

7−5

6

−2

8−4

6

f �1�x� � x � 5 f �1�x� � x � 7

t � 2.18

N�T�t��N�T�t�� 100t2 � 275�v � d��10� � 3330

0137

4000

(v + d)(t)

v(t)

d(t)

�v � d��t� � �36.04t2 � 804.6t � 1112f �x� � 3�x, g�x� � x � 2f �x� � x3, g�x� � 6x � 5

f, g, f � g, and g � f:

3�x3 � 3x � 3g � f :f, g, f � g,

x � 8x �83

x < 3x2 � 4�3 � x

;

�x2 � 4��3 � x

x2 � 4 � �3 � xx2 � 4 � �3 � x

x �12

x2 � 32x � 1

;

2x3 � x2 � 6x � 3

x2 � 2x � 4x2 � 2x � 2

h�x� �12 f �x� � 1

x

3

2

2

−2

−3

3

1

−2−3

h

y

f �x� � �x�h�x� � �2 f �x � 4�

x2 6 8

2

−2

−4

−6

−8

−2

h

y

f �x� � �xh�x� � �

13 f �x�

x

3

2

2 3−1−1

−2

−3

−2−3

1

h

y

f �x� � x3

333202CB01b_AN.qxd 4/13/06 5:17 PM


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d.

(Continued)

148. (a)

(b)

(c) The graphs are reflections of each other in the line


149. (a)(b)

(c) The graph of is the reflection of the graph of inthe line

(d) f has a domain of and a range of has a domain of and a range of

150. (a)(b)

(c) The graphs are reflections of each other in the line


151.

152.

153. (a)

(b) The model is a good fit for the actual data.154. (a) and (b)

This model is a good fit for the actual data.(c) 2008: $10,940.36(d) The factory sales of electronic gaming software in the

U.S. increases by $627.02 million dollars each year.155. Model: 3.2 kilometers, 16 kilometers156. 2438.7 kilowatts 157. A factor of 4158. 666 boxes 159.160. $44.80161. False. The graph is reflected in the -axis, shifted nine

units to the left, and then shifted 13 units downward.

162. True. If and then the domain of isall real numbers, which is equal to the range of and viceversa.

163. True. If is directly proportional to then soTherefore, is directly proportional to

164. The Vertical Line Test is used to determine if the graph ofis a function of The Horizontal Line Test is used to

determine if a function has an inverse function.165. A function from a set to a set is a relation that assigns

to each element in the set exactly one element in theset B.

yAxBA

x.y

y.xx � �1�k�y.y � kx,x,y

fgg�x� � 3�x,f �x� � x3

y

x

3

−3 3 6 9−12 −6−9−3

−6

−9

−12

−18

x

2 hours, 26 minutes

m �85k;

S � 627.02t � 346

52000

13

8000

y

t

Med

ian

inco

me

(in

thou

sand

s of

dol

lars

)

Year (5 ↔ 1995)

5 6 7 8 9 10 11 12

45

50

55

60

65

f �1�x� � x � 2, x ≥ 0x ≥ 2;

f �1�x� ��x2

� 4x ≥ 4;

f �1

y � x.

−4 −3 −2 1 3 4

−4

−3

−2

1

3

4

x

f

y

f −1

f �1�x� � 3 �x � 2

��1, ��.�0, ��f �1�0, ��;��1, ��

y � x.ff �1

–1 2 3 4 5–1

2

3

4

5

x

f −1

f

y

f �1�x� � x2 � 1, x ≥ 0

f �1

y � x.

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

2

4

6

8

x

f −1

f

y

f �1�x� �x � 7

5

333202CB01b_AN.qxd 4/13/06 5:17 PM

Precalculus with Limits, Answers to Chapter Test 51C

opyr

ight

©H

ough

ton

Mif

flin

Com

pany

. All

righ

ts r

eser

ved.

Chapter Test (page 123)

1.

Midpoint: ; Distance: 2.

3. No symmetry 4. axis symmetry

5. axis symmetry

6. 7.

8.

9. (a) (b)

10. (a) (b) (c)

11.12. (a)

(b)

(c) Increasing on Decreasing on

(d) Even

13. (a) 0, 3(b)


(d) Neither even nor odd14. (a)

(b)


(d) Neither even nor odd15.

16. Reflection in the -axis of

17. Reflection in the -axis, horizontal shift, and vertical shiftof

x6

2

−2−4−6 42

4

10

8

−2

y

y � �xx

y

x−2−4−6

−2

−4

−6

4

6

4 6

y � �x�x

x

10

2−2−10

4 6−6

−20

−30

20

30

y

��, �5��5, ��

−12 6

−2

10

�5

�2, 3��, 2�

−2 4

10

−10

��, �0.31�, �0, 0.31��0.31, 0�, �0.31, ��

−0.1

1−1

0.1

0, ±0.4314�10 ≤ x ≤ 10

�xx2 � 18x

�1

28�

18

7x � 4y � 53 � 04x � 7y � 44 � 0

17x � 10y � 59 � 0

2x � y � 1 � 0�x � 1�2 � �y � 3�2 � 16

−1−2−3−4 1 2 3 4

−2

−3

−4

1

2

3

4

y�

x�(−1, 0)

(0, −1)

(1, 0)

y-

−1−2−3−4 1 2 3 4−1

−2

1

2

3

4

5

6

y

x

(0, 4)

(−4, 0) (4, 0)−1−2−3−4 1 2 3 4−1

−2

−3

−4

1

2

3

4

y

x

(0, 3)

35

, 0( (

y-

11.937 centimeters�89�2, 52�

x

(−2, 5)

(6, 0)

5 64321

2

3

5

1

−1−2

−2

6

y

333202CB01b_AN.qxd 4/13/06 5:17 PM

(Continued)

18. (a) (b)(c)

(d)

(e)(f)

19. (a) (b)

(c) (d)

(e) (f)

20. 21. No inverse

22. 23.

24. 25. b �48a

A �256

xy

v � 6�sf �1�x� � �13 x�2�3, x ≥ 0

f �1�x� � 3�x � 8

2�xx

, x > 0�x2x

, x > 0

12x3�2, x > 0

2�xx

, x > 0

1 � 2x3�2

x, x > 0

1 � 2x3�2

x, x > 0

�9x 4 � 30x2 � 163x 4 � 24x3 � 18x2 � 120x � 68

3x2 � 7�x2 � 4x � 5

, x � �5, 1

�3x 4 � 12x3 � 22x2 � 28x � 354x2 � 4x � 122x2 � 4x � 2

Precalculus with Limits, Answers to Chapter Test 52

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333202CB01b_AN.qxd 4/13/06 5:17 PM

Precalculus with Limits, Answers to Problem Solving 53

Problem Solving (page 125)1. (a) (b)

(c)

Both jobs pay the same monthly salary if sales equal$15,000.

(d) No. Job 1 would pay $3400 and job 2 would pay $3300.

2. Mapping numbers onto letters is not a function since eachnumber corresponds to three letters.Mapping letters onto numbers is a function since every letter is assigned exactly one number.

3. (a) The function will be even.(b) The function will be odd.(c) The function will be neither even nor odd.

4.

Both graphs are already symmetric with respect to the line

General formula:

5.

6.

7. (a) hours (b) miles per hour

(c)

Domain:

Range:

(d)

8. (a) 1 (b) 1.5 (c) 1.75 (d) 1.875 (e) 1.9375(f) Yes, 2.(g) a.

b.c.d.e.

(h)

9. (a) (b)

(c)

(d)

(e)

(f) Answers will vary.(g)

10. (a)(b)(c)

(d)(e) The distance yields a time of 1.68 hours.

11. (a) (b)y

x−3

−3

−2

−1

1

2

3

−2 −1 1 2 3

y

x−3

−3

−1

1

2

3

−2 −1 1 2 3

x � 1x � 1

00 3

3

0 ≤ x ≤ 3T �

12�4 � x2 �

14�x2 � 6x � 10

� f � g��1�x� � �g�1� f �1��x�

�g�1� f �1��x� �

12

3�x � 1

f �1�x� � 3�x � 1; g�1�x� �12x;

� f � g��x� � 8x3 � 1; � f � g��1�x� �12

3�x � 1;

�g�1� f �1��x� �

14 x � 6

f �1�x� �14 x; g�1�x� � x � 6

� f � g��1�x� �14x � 6� f � g��x� � 4x � 24

y � 2x � 2y � 1.9375x � 1.9375y � 1.875x � 1.875y � 1.75x � 1.75y � 1.5x � 1.5y � x � 1

y

x30

500

1000

1500

2000

2500

3000

3500

4000

60 90 120 150

Dis

tanc

e (i

n m

iles)

Hours

0 ≤ y ≤ 3400

0 ≤ x ≤1190

9

y ��180

7x � 3400

2557812

3

f �x� � � 127 x �

167 ,

�127 x �

1287 ,

2.5 ≤ x ≤ 6

6 < x ≤ 9.5

�6, 8�

� f �x�� . . . � a2�� x�2 � a0

f ��x� � a2n�� x�2n � a2n�2�� x�2n�2

f �x� � a2n x2n � a2n�2 x2n�2 � . . . � a2 x2 � a0

c ≥ 0y � �x � c,y � x.

y

x

1

2

3

1 2 3

y

x−3

−3

−2

−1

1

2

3

−2 −1 1 2 3

g�x� � �xf �x� � x

00 30,000

(15,000, 3,050)

5,000

W2 � 2300 � 0.05SW1 � 2000 � 0.07S

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333202CB01b_AN.qxd 4/13/06 5:17 PM

(Continued)

(c) (d)

(e) (f)

12. (a) Domain: all real numbers x except Range: all real numbers y except

(b)

Domain: all real numbers x except

(c)The graph is not a line because there are holes at and

13. Proof

14. (a) (b)

(c) (d)

(e) (f)

(g)

15. (a)

(b)

(c)

(d)

y

x−4 −2−3 2 3 4

4

3

2

−2

−1

−3

−4

y

x−4 −2−3 −1 2 31 4

4

3

2

−2

−3

−4

y

x−4 −2−3 2 3 4

4

3

2

−2

1

−3

−4

y

x−4 −2−3 2 3 4

4

3

2

−2

−1

−3

−4

y

x−4 −2−3 2 3 4

4

1

−2

−1

−3

−4

y

x−3−4 −1 1 3 4

4

3

1

−2

−3

−4

y

x−3−4 −1 1 2 3 4

4

3

1

−2

−3

−4

x � 1.x � 0

f � f � f �x�� x

x � 0, 1

f � f �x�� x � 1

x

y � 0x � 1

y

x−3

−3

−2

−1

1

3

−2 −1 1 2 3

y

x−3

−3

−2

−1

1

2

3

−2 −1 1 2 3

y

x−3

−3

−2

−1

2

3

−2 −1 1 2 3

y

x−3

−3

−2

−1

1

2

3

−2 −1 1 2 3

Precalculus with Limits, Answers to Problem Solving 54

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d.

0 4

0 4�2�4f � f�1�x��

�2�4x

0 1

5 1 �5�3� f � f �1��x�

�2�3x

0 1

4 0 2 6� f � f �1��x�

�2�3x

0 4

2 1 1 3� f �1�x��3�4x

333202CB01b_AN.qxd 4/13/06 5:17 PM

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