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CHAPTER P Preparation for Calculus Section P.1 Graphs and Models . . . . . . . . . . . . . . . . . . . . . . 2 Section P.2 Linear Models and Rates of Change . . . . . . . . . . . . 11 Section P.3 Functions and Their Graphs . . . . . . . . . . . . . . . . . 22 Section P.4 Fitting Models to Data . . . . . . . . . . . . . . . . . . . . 31 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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Page 1: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

C H A P T E R PPreparation for Calculus

Section P.1 Graphs and Models . . . . . . . . . . . . . . . . . . . . . . 2

Section P.2 Linear Models and Rates of Change . . . . . . . . . . . . 11

Section P.3 Functions and Their Graphs . . . . . . . . . . . . . . . . . 22

Section P.4 Fitting Models to Data . . . . . . . . . . . . . . . . . . . . 31

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Page 2: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

7.

x

2

−4

−2

−6

6

−4−6 4 6

(−3, −5) (3, −5)

(−2, 0)

(0, 4)

(2, 0)

y

y � 4 � x2 8.

−6 −4 −2−2

2

2

4

4 6

6

8

10

y

x

y � �x � 3�2

x 0 2 3

y 0 4 0 �5�5

�2�3

C H A P T E R PPreparation for Calculus

Section P.1 Graphs and Models

2

1.

x-intercept:

y-intercept:

Matches graph (b).

�0, 2�

�4, 0�

y � �12 x � 2

3.

x-intercepts:

y-intercept:

Matches graph (a).

�0, 4�

�2, 0�, ��2, 0�

y � 4 � x2

5.

x2

2

−4

−6

−8

4

6

8

−4−6−8 4 6 8

(−4, −5)(−2, −2)

(0, 1)

(2, 4)

(4, 7)

y

y �32 x � 1

x 0 2 4

y 1 4 7�2�5

�2�4

2.

x-intercepts:

y-intercept:

Matches graph (d).

�0, 3�

��3, 0�, �3, 0�

y � �9 � x2

4.

x-intercepts:

y-intercept:

Matches graph (c).

�0, 0�

�0, 0�, ��1, 0�, �1, 0�

y � x3 � x

6.

−6 −4 −2 2

2

4

6

6

10

y

x

y � 6 � 2x

x 0 1 2 3 4

y 10 8 6 4 2 0 �2

�1�2

x 0 1 2 3 4 5 6

y 9 4 1 0 1 4 9

Page 3: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.1 Graphs and Models 3

9.

x

2

−2

4

6

−4−6 2

(−3, 1) (−1, 1)

(−4, 2)

(−2, 0)

(0, 2)(1, 3)

(−5, 3)

y

y � �x � 2�

11.

x

4

6

8

2

−6

−8

−4

−10

10

−2 2 1614 1812

(0, −4)

(1, −3)

(4, −2) (16, 0)

y

(9, −1)

y � �x � 4

x 0 1

y 3 2 1 0 1 2 3

�1�2�3�4�5

x 0 1 4 9 16

y 0�1�2�3�4

10.

−3 3

3

4

−2

−2

2

2

−1−1

1

1

y

x

y � �x� � 1

12.

−5 5 10 15 20

2

3

4

5

y

x

y � �x � 2

x 0 1 2 3

y 2 1 0 0 1 2�1

�1�2�3

x 0 2 7 14

y 0 1 2 3 4�2

�1�2

16.

Note that when or x � 10.x � 0y � 10

Xmin = -30Xmax = 30Xscl = 5Ymin = -10Ymax = 40Yscl = 5

15.

Note that when x � 0.y � 4

Xmin = -3Xmax = 5Xscl = 1Ymin = -3Ymax = 5Yscl = 1

13.

−1−2 1 2 3−1

1

2

3

x

(1, 2)

(2, 1)

(−2, −1)

(−1, −2)

−3, − ) 23) ) 2

33, )

y

y �2

x14.

x

−3, − ) 14)

−1, − ) 12) ) 1

45, ) ) 1

23, )

y

2 3 4 5

−3

−4

2

3

4

(2, 0)

(0, −1)1

y �1

x � 1

x 0 1 2 3

y Undef. 2 1 23�2�1�

23

�1�2�3 x 0 1 2 3 5

y Undef. 1 14

12�1�

12�

14

�1�3

Page 4: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

4 Chapter P Preparation for Calculus

18.

(a)

(b) and �x, �4� � �1, �4��x, �4� � ��1.65, �4�

��0.5, y� � ��0.5, 2.47�

− 9

−6

9

6y � x5 � 5x17.

(a)

(b) �x, 3� � ��4, 3� �3 � �5 � ��4� ��2, y� � �2, 1.73� �y � �5 � 2 � �3 � 1.73�

−6 6

−3

5

( 4.00, 3)−(2, 1.73)

y � �5 � x

19.

y-intercept:

x-intercepts:

x � �2, 1; ��2, 0�, �1, 0�

0 � �x � 2��x � 1�

0 � x2 � x � 2

y � �2; �0, �2�y � 02 � 0 � 2

y � x2 � x � 2

21.

y-intercept:

x-intercepts:

x � 0, ±5; �0, 0�; �±5, 0�

0 � x2��5 � x��5 � x�

0 � x2�25 � x2

y � 0; �0, 0�

y � 02�25 � 02

y � x2�25 � x2

20.

y-intercept:

x-intercepts:

x � 0, ±2; �0, 0�, �±2, 0�

0 � x�x � 2��x � 2�

0 � x3 � 4x

y � 0; �0, 0�

y2 � 03 � 4�0�

y 2 � x3 � 4x

22.

y-intercept:

x-intercept:

x � 1; �1, 0�

0 � �x � 1��x2 � 1

y � �1; �0, �1�

y � �0 � 1��02 � 1

y � �x � 1��x2 � 1

24.

y-intercept:

x-intercepts:

x � 0, �3; �0, 0�, ��3, 0�

0 �x�x � 3��3x � 1�2

0 �x2 � 3x

�3x � 1�2

y � 0; �0, 0�

y �02 � 3�0�

�3�0� � 1�2

y �x2 � 3x

�3x � 1�223.

y-intercept: None. x cannot equal 0.

x-intercept:

x � 4; �4, 0�

0 � 2 � �x

0 �3�2 � �x�

x

y �3�2 � �x�

x

25.

y-intercept:

x-intercept:

x � 0; �0, 0�

x2�0� � x2 � 4�0� � 0

y � 0; �0, 0�

02�y� � 02 � 4y � 0

x2y � x2 � 4y � 0

Page 5: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.1 Graphs and Models 5

26.

y-intercept:

x-intercept:

Note: is an extraneous solution.x � ��3�3

x ��33

; �33

, 0

x � ±�33

x2 �13

3x2 � 1

4x2 � x2 � 1

2x � �x2 � 1

0 � 2x � �x2 � 1

y � �1; �0, �1�

y � 2�0� � �02 � 1

y � 2x � �x2 � 1 27. Symmetric with respect to the y-axis since

y � ��x�2 � 2 � x2 � 2.

29. Symmetric with respect to the x-axis since

��y�2 � y2 � x3 � 4x.

31. Symmetric with respect to the origin since

��x���y� � xy � 4.

33.

No symmetry with respect to either axis or the origin.

y � 4 � �x � 3

35. Symmetric with respect to the origin since

y �x

x2 � 1.

�y ��x

��x�2 � 1

37. is symmetric with respect to the y-axis

since y � ���x�3 � ��x�� � ���x3 � x�� � �x3 � x�.y � �x3 � x�

28.

No symmetry with respect to either axis or the origin.

y � x2 � x

30. Symmetric with respect to the origin since

y � x3 � x.

�y � �x3 � x

��y� � ��x�3 � ��x�

32. Symmetric with respect to the x-axis since

x��y�2 � xy2 � �10.

34. Symmetric with respect to the origin since

xy � �4 � x2 � 0.

��x���y� � �4 � ��x�2 � 0

36. is symmetric with respect to the y-axis

since y ���x�2

��x�2 � 1�

x2

x2 � 1.

y �x2

x2 � 1

38. is symmetric with respect to the x-axis

since

�y� � x � 3.

��y� � x � 3

�y� � x � 3 39.

Intercepts:

Symmetry: none

�23 , 0�, �0, 2�

0

y

)20( ,2

1

x321

1

,23

y � �3x � 2

Page 6: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

6 Chapter P Preparation for Calculus

40.

Intercepts:

Symmetry: none

1 2 3 4

−1

−2

1

3

(4, 0)

(0, 2)

y

x

�4, 0�, �0, 2�

y � �x2

� 2 41.

Intercepts:

Symmetry: none

x108

)0,8(

y

42

)4,

2

(

2

0

2

8

6

10

�8, 0�, �0, �4�

y �12

x � 4 42.

Intercepts:

Symmetry: none

−1

−1

1 2

2

−2

y

x

32

, 0( )(0, 1)

�0, 1�, ��32, 0�

y �23

x � 1

43.

Intercepts:

Symmetry: y-axis

x)0(1,

0, 1)

y

2

),( 1 0

(

1

2

2 2−

�1, 0�, ��1, 0�, �0, 1�

y � 1 � x2 44.

Intercept:

Symmetry: y-axis

3 6−3−6

9

12

(0, 3)

y

x

�0, 3�

y � x2 � 3 45.

Intercepts:

Symmetry: none

x

2

−2

8

10

12

−8 −6−10 2 4(−3, 0)

(0, 9)

y

��3, 0�, �0, 9�

y � �x � 3�2

46.

Intercepts:

Symmetry: none

1 2 3−1−2−3

1

3

5

2

4

(0, 0)12

− , 0( )

y

x

�0, 0�, ��12 , 0�

y � 2x2 � x � x�2x � 1� 47.

Intercepts:

Symmetry: none

y

5

4

3

3

x32

,

1

1

1

,,

23

)02 )(0 2(

�� 3�2, 0�, �0, 2�

y � x3 � 2 48.

Intercepts:

Symmetry: origin

−3

−3

−1−1

−2

1 3

3

(−2, 0) (2, 0)(0, 0)

y

x

�0, 0�, �2, 0�, ��2, 0�

y � x3 � 4x

49.

Intercepts:

Symmetry: none

Domain: x ≥ �2

�0, 0�, ��2, 0�

x

1

2

3

4

5

−2

6

−4 −3 −1 41 2 3

(−2, 0) (0, 0)

yy � x�x � 2 50.

Intercepts:

Symmetry: y-axis

Domain: ��3, 3�

��3, 0�, �3, 0�, �0, 3�

−2−4 2 4

−2

2

4

6

(0, 3)

(3, 0)( 3,0)−

5

1

−1−1

1

y

x

y � �9 � x2

Page 7: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.1 Graphs and Models 7

51.

Intercept:

Symmetry: origin

x1

−2

−3

−4

2

3

4

−2 −1−3−4 2 3 4

(0, 0)

y

�0, 0�

x � y3 52.

Intercepts:

Symmetry: x-axis

1

−3

3

−1−2−5

( 4, 0)−(0, 2)

(0, 2)−

y

x

�0, 2�, �0, �2�, ��4, 0�

x � y2 � 4 53.

Intercepts: none

Symmetry: originy

3

1

2

x321

y �1x

54.

Intercept:

Symmetry: y-axis

−6 −4 −2

2

2

12

10

4 6

(0, 10)

y

x

�0, 10�

y �10

x2 � 155.

Intercepts:

Symmetry: y-axis

x2

2

−4

−2

−6

−8

4

6

8

−4 −2−8 4 6 8

(−6, 0)

(0, 6)

(6, 0)

y

�0, 6�, ��6, 0�, �6, 0�

y � 6 � �x� 56.

Intercepts:

Symmetry: none

2

2

4

4

6 8

8

y

x

(0, 6)

(6, 0)

�0, 6�, �6, 0�

y � �6 � x�

57.

Intercepts:

Symmetry: x-axis

1

−4

−11

4

(−9, 0)

(0, −3)

(0, 3)

�0, 3�, �0, �3�, ��9, 0�

y � ±�x � 9

y2 � x � 9

y2 � x � 9 58.

Intercepts:

Symmetry: origin and both axes

Domain:

−3 3

−2

(0, 1)

(0, 1)−

(2, 0)( 2, 0)−

2

��2, 2�

��2, 0�, �2, 0�, �0, �1�, �0, 1�

x2 � 4y2 � 4 ⇒ y � ±�4 � x2

259.

Intercepts:

Symmetry: x-axis

8

−3

−1

3

(6, 0)

( )0, 2

( )0, − 2

�6, 0�, �0, �2�, �0, ��2�

y � ±�2 �x3

3y2 � 6 � x

x � 3y2 � 6

60.

Intercept:

Symmetry: x-axis

�83, 0�

y � ±�34 x � 2

4y2 � 3x � 8−6

−6

6

12

83

, 0( )

3x � 4y2 � 8 61.

The corresponding y-value is

Point of intersection: �1, 1�y � 1.

1 � x

3 � 3x

2 � x � 2x � 1

2x � y � 1 ⇒ y � 2x � 1

x � y � 2 ⇒ y � 2 � x

Page 8: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

8 Chapter P Preparation for Calculus

62.

The corresponding y-value is

Point of intersection: �2, �3�

y � �3.

x � 2

7x � 14

2x � 13 � 1 � 5x

2x � 133

�1 � 5x

3

5x � 3y � 1 ⇒ y �1 � 5x

3

2x � 3y � 13 ⇒ y �2x � 13

363.

The corresponding y-values are (for )and (for ).

Points of intersection: �2, 2�, ��1, 5�

x � �1y � 5x � 2y � 2

x � 2, �1

0 � �x � 2��x � 1�

0 � x2 � x � 2

6 � x2 � 4 � x

x � y � 4 ⇒ y � 4 � x

x2 � y � 6 ⇒ y � 6 � x2

64.

The corresponding y-values are and

Points of intersection: ��1, �2�, �2, 1�

y � 1.y � �2

x � �1 or x � 2

0 � x2 � x � 2 � �x � 1��x � 2�

3 � x � x2 � 2x � 1

3 � x � �x � 1�2

y � x � 1

x � 3 � y2 ⇒ y2 � 3 � x 65.

The corresponding y-values are and

Points of intersection: ��1, �2�, �2, 1�

y � 1.y � �2

x � �1 or x � 2

0 � 2x2 � 2x � 4 � 2�x � 1��x � 2�

5 � x2 � x2 � 2x � 1

5 � x2 � �x � 1�2

x � y � 1 ⇒ y � x � 1

x2 � y 2 � 5 ⇒ y2 � 5 � x 2

66.

The corresponding y-values are and

Points of intersection: �3, 4�, �5, 0�

y � 0.y � 4

x � 3 or x � 5

0 � 5x2 � 40x � 75 � 5�x � 3��x � 5�

25 � x2 � 100 � 40x � 4x2

25 � x2 � �10 � 2x�2

2x � y � 10 ⇒ y � 10 � 2x

x2 � y2 � 25 ⇒ y2 � 25 � x2 67.

The corresponding y-values are and

Points of intersection: �0, 0�, ��1, �1�, �1, 1�

y � 1.y � 0, y � �1,

x � 0, x � �1, or x � 1

x�x � 1��x � 1� � 0

x3 � x � 0

x3 � x

y � x

y � x3

68.

The corresponding y-values are and

Points of intersection: �1, �3�, ��2, 0�

y � 0.y � �3

x � 1 or x � �2

�x � 1�2�x � 2� � 0

x3 � 3x � 2 � 0

x3 � 4x � ��x � 2�

y � ��x � 2� y � x3 � 4x 69.

��1, �5�, �0, �1�, �2, 1�

x � �1, 0, 2

x�x � 2��x � 1� � 0

x3 � x2 � 2x � 0

x3 � 2x2 � x � 1 � �x2 � 3x � 1

y � �x2 � 3x � 1

y � x3 � 2x2 � x � 1

−4 6

−8

4

y x x= + 3 1− −2

y x x x= 2 + 1− −3 2

(2, 1)

(0, 1)−

( 1, 5)− −

Page 9: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.1 Graphs and Models 9

70.

−3 3

−2

2

(1, 0)( 1, 0)−

(0, 1)

y x= −1 2

y x x= − +2 124

��1, 0�, �0, 1�, �1, 0�

x � �1, 0, 1

0 � x2�x � 1��x � 1�

0 � x 4 � x2

1 � x2 � x 4 � 2x2 � 1

y � 1 � x2

y � x 4 � 2x2 � 1 71.

Points of intersection:

Analytically,

x � �2, y � 2 ⇒ ��2, 2�. x � �3, y � �3 ⇒ ��3, �3�

�x � 3��x � 2� � 0

x2 � 5x � 6 � 0

x � 6 � �x2 � 4x

�x � 6 � ��x2 � 4x

��3, �3� � ��3, 1.732���2, 2�,

−7 2

4

−2

x + 6y =

−x2 − 4x

(−2, 2)

y =

(3, )3

y � ��x2 � 4x

y � �x � 6

72. Points of intersection:

Analytically,

or

or

Hence, �1, 5�.�3, 3�,

x � 1. x � 3

2x � 3 � �x 2x � 3 � x

�2x � 3� � x

−4 8

7

−1

y = 6 − x

y = − 2x - 3 + 6

(1, 5)

(3, 3)

��2x � 3� � 6 � 6 � xy � 6 � x

�1, 5��3, 3�,y � ��2x � 3� � 6

73. (a) Using a graphing utility, you obtain

(b)

(c) For 2010, and y � 217.t � 40

35

−50

−5

250

y � �0.007t2 � 4.82t � 35.4.

74. (a)

(b)

(c) For 2010, andacres.� 405y � �0.13�602� � 11.1�60� � 207

t � 60

0 50

500

0

y � �0.13t2 � 11.1t � 207

75.

The other root, does not satisfy the equation

This problem can also be solved by using a graphing utility and finding the intersection of the graphs of C and R.

R � C.x � 2949,

x � 3133 units

0 � 10.8241x2 � 65,830.25x � 100,000,000 Use the Quadratic Formula.

30.25x � 10.8241x2 � 65,800x � 100,000,000

�5.5�x �2� �3.29x � 10,000�2

5.5�x � 10,000 � 3.29x C � R

Page 10: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

10 Chapter P Preparation for Calculus

76.

If the diameter is doubled, the resistance is changed by approximately a factor of For instance, andy�40� � 6.36125.

y�20� � 26.55514.

00

400

100

y �10,770

x2 � 0.37

77. (other answers possible)y � �x � 2��x � 4��x � 6� 78. (other answers possible)y � �x �52��x � 2��x �

32�

80. (a) If is on the graph, then so is by y-axissymmetry. Since is on the graph, then so is

by x-axis symmetry. Hence, the graph issymmetric with respect to the origin. The converse isnot true. For example, has origin symmetry but is not symmetric with respect to either the x-axisor the y-axis.

(b) Assume that the graph has x-axis and origin symmetry.If is on the graph, so is by x-axis symmetry. Since is on the graph, then so is

by origin symmetry.Therefore, the graph is symmetric with respect to the y-axis. The argument is similar for y-axis and origin symmetry.

��x, ���y�� � ��x, y��x, �y�

�x, �y��x, y�

y � x3

��x, �y���x, y�

��x, y��x, y�79. (i) matches (b).

Use

(ii) matches (d).

Use

(iii) matches (a).

Use

(iv) matches (c).

Use �1��36� � k ⇒ k � 36, thus, xy � 36.�1, 36�:

xy � k

3 � k�1�3�2 ⇒ k � 3, thus, y � 3x3�2.�1, 3�:

y � kx3�2

�9 � �1�2 � k ⇒ k � �10, thus, y � x2 � 10.

�1, �9�:

y � x2 � k

7 � k�1� � 5 ⇒ k � 2, thus, y � 2x � 5.�1, 7�:

y � kx � 5

85.

Circle of radius 2 and center �0, 4�

x2 � �y � 4�2 � 4

x2 � y2 � 8y � 12 � 0

3x2 � 3y2 � 24y � 36 � 0

4x2 � 4y2 � 24y � 36 � x2 � y2

4�x2 � �y � 3�2� � x2 � y2

x

y

−1−2 2 3

−1

1

2

4

(0, 3)

(x, y)

(0, 0)

2��x � 0�2 � �y � 3�2 � ��x � 0�2 � �y � 0�2

86. Distance from the origin Distance from

Note: This is the equation of a circle!

�1 � K 2�x 2 � �1 � K 2�y 2 � 4K 2x � 4K 2 � 0

x2 � y 2 � K 2�x2 � 4x � 4 � y2�

�x2 � y2 � K��x � 2�2 � y2, K � 1

�2, 0�� K �

82. True 83. True; the x-intercepts are

�b ± �b2 � 4ac2a

, 0.

84. True; the x-intercept is

�b

2a, 0.

81. False; x-axis symmetry means that if is on the graph, then is also on the graph.�1, 2��1, �2�

Page 11: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.2 Linear Models and Rates of Change

Section P.2 Linear Models and Rates of Change 11

1. m � 1 2. m � 2 3. m � 0

4. m � �1 5. m � �12 6. m �403

7.

2

3

4

5

−1

1

x4 51 3

(2, 3)

m = −2

m = 1

32

m = −m isundefined.

y 8.

2

4

6

−2

−2−6

m = 0

m =

m = 3m = 3−

( 4, 1)−

13

3

1−1−5

y

x

9.

x

y

1 2 3 5 6 7−1

(3, 4)−

(5, 2)

−2

−3

−4

−5

1

2

3

�62

� 3

m �2 � ��4�

5 � 3

10.

4),( 2 4

), 2

2

(

1

12

1

12

y

x

� �23

m �4 � 2

�2 � 111.

undefined

x

1

2

−1

−2

3

4

5

6

−1−2 1 3 4 5 6

(2, 1)

(2, 5)

y

�40

m �5 � 12 � 2

12.

1

1

−1

−3

2

2

3 4 5

y

x

(3, −2) (4, −2)

m ��2 � ��2�

4 � 3� 0

13.

�1�21�4

� 2

x

−1

−2

−3

2

3

−2−3 21 3

y

( )12

23,− ( )3

416,−

m �2�3 � 1�6

�1�2 � ��3�4� 14.

�1

�3�8� �

83

1−1−2

1

2

3

−154

14

−( ),

78

34( ),

y

x

m ��3�4� � ��1�4��7�8� � �5�4�

15. Since the slope is 0, the line is horizontal and its equationis Therefore, three additional points are

and �3, 1�.�1, 1�,�0, 1�,y � 1.

16. Since the slope is undefined, the line is vertical and itsequation is Therefore, three additional points are , and ��3, 5�.��3, 3�,��3, 2�

x � �3.

17. The equation of this line is

.

Therefore, three additional points are and �3, 1�.

�2, 4�,�0, 10�,

y � �3x � 10

y � 7 � �3�x � 1�

18. The equation of this line is

.

Therefore, three additional points are and �0, 2�.

��1, 0�,��3, �4�,

y � 2x � 2

y � 2 � 2�x � 2�

Page 12: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

12 Chapter P Preparation for Calculus

19. (a)

(b)

By the Pythagorean Theorem,

x � 10�10 � 31.623 feet.

x2 � 302 � 102 � 1000

30 ft

10 ftx

Slope ��y�x

�13

20. (a) indicates that the revenues increase by 400 in one day.

(b) indicates that the revenues increase by 100 in one day.

(c) indicates that the revenues do not change from one day to the next.m � 0

m � 100

m � 400

21. (a)

(b) The slopes of the line segments are:

The population increased least rapidly between 2000 and 2001.

285.0 � 282.311 � 10

� 2.7

282.3 � 279.310 � 9

� 3.0

279.3 � 276.19 � 8

� 3.2

276.1 � 272.98 � 7

� 3.2

272.9 � 269.77 � 6

� 3.2

tPopu

latio

n (i

n m

illio

ns)

Year (6 ↔ 1996)

6 7 8 9 10 11

260

270

280

290

y 22. (a)

(b) The slopes are:

The rate changed most rapidly between 20 and 25 seconds. The change is �4.6 mph�sec.

43 � 6130 � 25

� �3.6

61 � 8425 � 20

� �4.6

84 � 8520 � 15

� �0.2

85 � 7415 � 10

� 2.2

74 � 5710 � 5

� 3.4

t

r

5 10 15 20 25 30

20

40

60

80

100

23.

Therefore, the slope is and the y-intercept is�0, 4�.

m � �15

y � �15 x � 4

x � 5y � 20 24.

Therefore, the slope is and the y-intercept is�0, �3�.

m �65

y �65 x � 3

6x � 5y � 15

25.

The line is vertical. Therefore, the slope is undefined andthere is no y-intercept.

x � 4 26.

The line is horizontal. Therefore, the slope is andthe y-intercept is �0, �1�.

m � 0

y � �1

27.

0 � 3x � 4y � 12

4y � 3x � 12

x

y

1−1−2−3−4

1

2

4

5

(0, 3)

y �34 x � 3 28.

x � 1 � 0

1−2−3

−1

1

2

3

( 1, 2)−

y

x

x � �1

Page 13: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.2 Linear Models and Rates of Change 13

29.

x

y

1 2 3 4

−1

2

3

4

(0, 0)

2x � 3y � 0

3y � 2x

y �23 x 30.

1 2−1−2−3

1

2

3

5

(0, 4)

y

x

y � 4 � 0

y � 4 31.

x

1

2

−3

−1

−2

−4

−5

3

−1−2 1 2 3 4 5 6

(3, −2)

y

y � 3x � 11 � 0

y � 3x � 11

y � 2 � 3x � 9

y � 2 � 3�x � 3�

32.

1 2−1−2−3

1

2

4

5

( 2, 4)−

y

x

3x � 5y � 14 � 0

5y � 20 � �3x � 6

y � 4 � �35

�x � 2� 33.

x2

2

−8

4

6

8

−4 −2−6−8 4 6 8

(2, 6)

(0, 0)

y

y � 3x

y � 0 � 3�x � 0�

m �6 � 02 � 0

� 3 34.

33)1,(

), 0

1

(

1

0

2 13

y

x

3x � y � 0

y � �3x

y � 0 � �3�x � 0�

m �3 � 0

�1 � 0� �3

35.

x

y

−2 −1 2 3 4 5−1

−2

−3

−5

1

2

(2, 1)

(0, 3)−

0 � 2x � y � 3

y � 1 � 2x � 4

y � 1 � 2�x � 2�

m �1 � ��3�

2 � 0� 2 36.

2− 4−6

−2

− 4

2

4 (1, 4)

( 3, 4)− −

1 3−3−5

−3

−5

3

5

−7

y

x

0 � 2x � y � 2

y � 4 � 2x � 2

y � 4 � 2�x � 1�

m �4 � ��4�1 � ��3� �

84

� 2 37.

12345

−2

6789

x−1 4 6 7 8 91 2 3

(5, 0)

(2, 8)

y

3y � 8x � 40 � 0

y � �83

x �403

y � 0 � �83

�x � 5�

m �8 � 02 � 5

� �83

38.

x � y � 3 � 0

y � 2 � �x � 1

y � 2 � �1�x � 1�

1 2 3−1−3 −2−4

1

3

2

5

7

6

(1, 2)

( 3, 6)−

y

x

m �6 � 2

�3 � 1�

4�4

� �1 39. Undefined

Vertical line x � 5

12345

−2

6789

x−1 4 6 7 8 91 2 3

(5, 1)

(5, 8)

y

m �8 � 15 � 5

Page 14: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

14 Chapter P Preparation for Calculus

40.

1

2)( ,3),(1

1

4

32

1 1 2 3 4

y

x

y � 2 � 0

y � �2

m � 0 41.

x1

2

1

3

4

−2 −1−3−4 2 3 4

( )12

72,

( )340,

y

22x � 4y � 3 � 0

y �112

x �34

y �34

�112

�x � 0�

m �7�2 � 3�41�2 � 0

�11�41�2

�112

42.

1−1−2

1

2

3

−154

14

−( ),

78

34( ),

y

x

32x � 12y � 37 � 0

12y � 3 � �32x � 40

y �14

��83 �x �

54�

�1

�3�8� �

83

m ��3�4� � ��1�4��7�8� � �5�4�

45.

3x � 2y � 6 � 0

x2

�y3

� 1 46.

3x � y � 2 � 0

3x � y � �2

�3x

2�

y2

� 1

x�2�3

�y

�2� 1 47.

x � y � 3 � 0

a � 3 ⇒ x � y � 3

3a

� 1

1a

�2a

� 1

xa

�ya

� 1

43.

x � 3 � 0

1 2 4

−1

−2

1

2

(3, 0)

y

x

x � 3 44.

xa

�yb

� 1

ba

x � y � b

y ��ba

x � b

( , 0)a

(0, )b

y

x

m � �ba

48.

x � y � 1 � 0

a � 1 ⇒ x � y � 1

1a

� 1

�3a

�4a

� 1

xa

�ya

� 1 49.

x

y

−1−2−3 1 2 3 4 5

−2

−4

−5

−6

1

2

y � 3 � 0

y � �3 50.

1 2 3 5

−1

−2

1

2

3

y

x

x � 4 � 0

x � 4

Page 15: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.2 Linear Models and Rates of Change 15

51.

x

y

−2 −1 1 2

−1

3

y � �2x � 1 52.

−3

−3

−4

3−2

−2

2

1

−1

y

x

(0, −1)

3y � x � 3 � 0

y �13 x � 1 53.

x1

2

1

3

4

−2

−2

−3

−4

−3−4 2 3 4

y

2y � 3x � 1 � 0

y �32 x �

12

y � 2 �32�x � 1�

54.

4 8−8−12−16−4

−8

12

16

y

x

y � 3x � 13

y � 1 � 3�x � 4� 55.

y

1

x32

1

2

3

12

y � 2x � 3

2x � y � 3 � 0 56.

−10 −8 −6 −2

−4

−6

2

4

y

x

y � �12 x � 3

x � 2y � 6 � 0

57. (a)

The lines do notappear perpendicular.10

−10

−10

10 (b)

The lines appearperpendicular.15

−10

−15

10

The lines are perpendicular because their slopes 1 and are negative reciprocals of each other.You must use a square setting in order for perpendicular lines to appear perpendicular. Answers depend on calculator used.

�1

58. (a)

The lines do notappear perpendicular.−5

−5

5

5

(b)

The lines appearperpendicular.−6

−4

4

6

The lines are perpendicular because their slopes 2 and are negative reciprocals of each other.You must use a square setting in order for perpendicular lines to appear perpendicular. Answers depend on calculator used.

�12

59.

(a)

(b)

x � 2y � 4 � 0

2y � 2 � �x � 2

y � 1 � �12 �x � 2�

2x � y � 3 � 0

y � 1 � 2x � 4

y � 1 � 2�x � 2�

m � 2

y � 2x �32

4x � 2y � 3 60.

(a)

(b)

x � y � 5 � 0

y � 2 � x � 3

y � 2 � 1�x � 3�

x � y � 1 � 0

y � 2 � �x � 3

y � 2 � �1�x � 3�

m � �1

y � �x � 7

x � y � 7

Page 16: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

16 Chapter P Preparation for Calculus

61.

(a)

(b)

40y � 24x � 53 � 0

40y � 35 � �24x � 18

y �78 � �

35�x �

34�

24y � 40x � 9 � 0

24y � 21 � 40x � 30

y �78 �

53�x �

34�

m �53

y �53x

5x � 3y � 0 62.

(a)

(b)

4x � 3y � 36 � 0

3y � 12 � 4x � 24

y � 4 �43�x � 6�

3x � 4y � 2 � 0

4y � 16 � �3x � 18

y � 4 � �34 �x � 6�

m � �34

y � �34 x �

74

3x � 4y � 7

63. The given line is vertical.

(a)

(b) y � 5 ⇒ y � 5 � 0

x � 2 ⇒ x � 2 � 0

64. (a)

(b) x � �1 ⇒ x � 1 � 0

y � 0

65. The slope is 125. when

V � 125�t � 4� � 2540 � 125t � 2040

t � 4. V � 2540 66. The slope is 4.5. when

V � 4.5�t � 4� � 156 � 4.5t � 138

t � 4.V � 156

67. The slope is when

V � �2000�t � 4� � 20,400 � �2000t � 28,400

t � 4.V � 20,400�2000. 68. The slope is when

� �5600t � 267,400

V � �5600�t � 4� � 245,000

t � 4.V � 245,000�5600.

69.

You can use the graphing utility to determine that the points of intersection are and Analytically,

The slope of the line joining and is Hence, an equation

of the line is

y � 2x.

y � 0 � 2�x � 0�

m � �4 � 0���2 � 0� � 2.�2, 4��0, 0�

x � 2 ⇒ y � 4 ⇒ �2, 4�.

x � 0 ⇒ y � 0 ⇒ �0, 0�2x�x � 2� � 0

2x2 � 4x � 0

x2 � 4x � x2

�2, 4�.�0, 0�

−1

−3 6(0, 0)

(2, 4)

70.

You can use the graphing utility to determine that the points of intersection are and Analytically,

The slope of the line joining and is Hence, an equation

of the line is

y � �x � 3.

y � 3 � �1�x � 0�

m � �0 � 3���3 � 0� � �1.�3, 0��0, 3�

x � 3 ⇒ y � 0 ⇒ �3, 0�.

x � 0 ⇒ y � 3 ⇒ �0, 3� 2x�x � 3� � 0

2x2 � 6x � 0

x2 � 4x � 3 � �x2 � 2x � 3

�3, 0�.�0, 3�

−9 9

−6

6

(0, 3)(3, 0)

y � �x2 � 2x � 3y � x2 � 4x � 3,

Page 17: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.2 Linear Models and Rates of Change 17

71.

The points are not collinear.

m1 � m2

m2 ��2 � 0

2 � ��1� � �23

m1 �1 � 0

�2 � ��1� � �1 72.

The points are not collinear.

m1 � m2

m2 �11 � 4�5 � 0

� �75

m1 ��6 � 47 � 0

� �107

73. Equations of perpendicular bisectors:

Setting the right-hand sides of the two equations equal and solving for x yields

Letting in either equation gives the point of intersection:

This point lies on the third perpendicular bisector, x � 0.

�0, �a2 � b2 � c2

2c �.

x � 0

x � 0.

y �c2

�a � b

�c �x �b � a

2 � y �

c2

�a � b

c �x �a � b

2 �( , )b c

( , 0)a( , 0)−a

a b+2

b a−2 c

2

c2 ( )( ) ,

,

y

x

74. Equations of medians:

Solving simultaneously, the point of intersection is

( , )b c

( , 0)a( , 0)−a

a b+2

c2( ),

b a−2

c2( ),

(0, 0)

y

x

�b3

, c3�.

y �c

�3a � b�x � a�

y �c

3a � b�x � a�

y �cb

x

75. Equations of altitudes:

Solving simultaneously, the point of intersection is

( , )b c

( , 0)a

( , 0)−a

y

x

�b, a2 � b2

c �.

y � �a � b

c�x � a�

x � b

y �a � b

c�x � a�

76. The slope of the line segment from to is:

The slope of the line segment from to is:

Therefore, the points are collinear.

m1 � m2

m 2 ����a2 � b2 � c2���2c� � �c�3�

0 � �b�3� ���3a2 � 3b2 � 3c2 � 2c2���6c�

�b�3�

3a2 � 3b2 � c2

2bc

�0, �a2 � b2 � c2

2c ��b3

, c3�

m1 ���a2 � b2��c � �c�3�

b � �b�3� ��3a2 � 3b2 � c2���3c�

�2b��3�

3a2 � 3b2 � c2

2bc

�b, a2 � b2

c ��b3

, c3�

Page 18: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

18 Chapter P Preparation for Calculus

77. Find the equation of the line through the points and

or

For .F � 72�, C � 22.2�

5F � 9C � 160 � 0

C �19�5F � 160�

F �95 C � 32

F � 32 �95 �C � 0�

m �180100 �

95

�100, 212�.�0, 32� 78.

If x � 137, C � 0.34�137� � 150 � $196.58.

C � 0.34x � 150

80. (a) Depreciation per year:

where 0 ≤ x ≤ 5.

y � 875 � 175x

8755 � $175

00

6

1000 (b)

(c)

x � 3.86 years

175x � 675

200 � 875 � 175x

y � 875 � 175�2� � $525

81. (a) Two points are and The slope is

(b)

If

(c) If p � 595, x �1

15�1330 � 595� � 49 units.

p � 655, x �1

15�1330 � 655� � 45 units.

15000

0

50

or x �1

15�1330 � p�

p � �15x � 750 � 580 � �15x � 1330

p � 580 � �15�x � 50�

m �625 � 58047 � 50

� �15.

�47, 625�.�50, 580� 82. (a)

(b)

(c) If .

(d) The slope shows the average increase in exam scorefor each unit increase in quiz score.

(e) The points would shift vertically upward 4 units. Thenew regression line would have a y-intercept 4 greaterthan before: y � 22.91 � 3.97x.

x � 17, y � 18.91 � 3.97�17� � 86.4

00

20

100

�x � quiz score, y � test score�y � 18.91 � 3.97x

79. (a)

(b) Using a graphing utility, the point of intersection is Analytically,

(c) Both jobs pay $17 per hour if 6 units are produced. For someone who can produce more than 6 units per hour, the secondoffer would pay more. For a worker who produces less than 6 units per hour, the first offer pays more.

y � 0.75�6� � 12.50 � 17.

3.3 � 0.55x ⇒ x � 6

0.75x � 12.50 � 1.30x � 9.20

�6, 17�.

00

30

50

(6, 17)

W2 � 1.30x � 9.20

W1 � 0.75x � 12.50

Page 19: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.2 Linear Models and Rates of Change 19

83. The tangent line is perpendicular to the line joining thepoint and the center

Slope of the line joining and is

The equation of the tangent line is

12y � 5x � 169 � 0.

y ��512

x �16912

y � 12 ��512

�x � 5�

125

.

�0, 0��5, 12�

x

y

(5, 12)

(0, 0)

4

−4 8 16−8

−8

−16

8

�0, 0�.�5, 12�84. The tangent line is perpendicular to the line joining the

point and the center of the circle,

Slope of the line joining and is

Tangent line:

4y � 3x � 24 � 0

y �34

x � 6

y � 3 �34

�x � 4�

1 � 31 � 4

��43

.

�4, �3��1, 1�

x

y

−2−6 2 4−2

−6

2

4

(1, 1)

(4, −3)

�1, 1�.�4, �3�

85. 4x � 3y � 10 � 0 ⇒ d � 4�0� � 3�0� � 10�42 � 32

�105

� 2 86. 4x � 3y � 10 � 0 ⇒ d � 4�2� � 3�3� � 10�42 � 32

�75

87. x � y � 2 � 0 ⇒ d � 1��2� � ��1��1� � 2�12 � 12

�5�2

�5�2

2

89. A point on the line is The distance from the point to is

d � 1�0� � 1�1� � 5�12 � 12

�1 � 5�2

�4�2

� 2�2.

x � y � 5 � 0�0, 1��0, 1�.x � y � 1

88. x � 1 � 0 ⇒ d � 1�6� � �0��2� � 1�12 � 02

� 7

90. A point on the line is The distance from the point to is

d � 3��1� � 4��1� � 10�32 � ��4�2

� �3 � 4 � 105

�95

.

3x � 4y � 10 � 0��1, �1���1, �1�.3x � 4y � 1

91. If then is the horizontal line The distance to is

If then is the vertical line The distance to is

(Note that A and B cannot both be zero.)

—CONTINUED—

d � x1 � ��CA � �

Ax1 � CA �

Ax1 � By1 � C�A2 � B2

.

�x1, y1�x � �C�A.Ax � C � 0B � 0,

d � y1 � ��CB � �

By1 � CB �

Ax1 � By1 � C�A2 � B2

.

�x1, y1�y � �C�B.By � C � 0A � 0,

Page 20: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

20 Chapter P Preparation for Calculus

91. —CONTINUED—

The slope of the line is The equation of the line through perpendicularto is:

The point of intersection of these two lines is:

(1)

(2)

(By adding equations (1) and (2))

(3)

(4)

(By adding equations (3) and (4))

point of intersection

The distance between and this point gives us the distance between and the line

� Ax1 � By1 � C�A2 � B2

���A2 � B2��C � Ax1 � By1�2

�A2 � B2�2

����A�C � By1 � Ax1�A2 � B2 �2

� ��B�C � Ax1 � By1�A2 � B2 �2

����AC � ABy1 � A2x1

A2 � B2 �2� ��BC � ABx1 � B2y1

A2 � B2 �2

d ����AC � B2x1 � ABy1

A2 � B2 � x1�2� ��BC � ABx1 � A2y1

A2 � B2 � y1�2

Ax � By � C � 0.�x1, y1��x1, y1�

��AC � B2x1 � ABy1

A2 � B2 , �BC � ABx1 � A2y1

A2 � B2 �

y ��BC � ABx1 � A2y1

A2 � B2

�A2 � B2�y � �BC � ABx1 � A2y1

Bx � Ay � Bx1 � Ay1⇒ �ABx � A2 y � �ABx1 � A2 y1

Ax � By � �C ⇒ ABx � B2y � �BC

x ��AC � B2x1 � ABy1

A2 � B2

�A2 � B2�x � �AC � B2x1 � ABy1

Bx � Ay � Bx1 � Ay1 ⇒ B2x � ABy � B2x1 � ABy1

Ax � By � �C ⇒ A2x � ABy � �AC

Bx1 � Ay1 � Bx � Ay

Ay � Ay1 � Bx � Bx1

y � y1 �BA

�x � x1�

Ax � By � C � 0�x1, y1��A�B.Ax � By � C � 0

92.

The distance is 0 when In this case, the line contains the point �3, 1�.y � �x � 4m � �1.

� 3m � 3�m2 � 1

d � Ax1 � By1 � C�A2 � B2

� m3 � ��1��1� � 4�m2 � ��1�2 − 9 9

−4

( 1, 0)−

8y � mx � 4 ⇒ mx � ��1�y � 4 � 0

Page 21: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.2 Linear Models and Rates of Change 21

93. For simplicity, let the vertices of the rhombus be and as shown in the figure. The

slopes of the diagonals are then

and

Since the sides of the rhombus are equal,and we have

Therefore, the diagonals are perpendicular.

m1m2 �c

a � b�

cb � a

�c2

b2 � a2 �c2

�c2 � �1.

a2 � b2 � c2,

m2 �c

b � a.m1 �

ca � b

�a � b, c�,�b, c�,�a, 0�,

( , 0)a

( , )a b c+( , )b c

(0, 0)

y

x

�0, 0�,

94. For simplicity, let the vertices of the quadrilateral be and as shown in the figure.

The midpoints of the sides are

and

The slope of the opposite sides are equal:

Therefore, the figure is a paralleogram.

0 �e2

a2

�d2

c2

�c � e

2a � b

2�

b � d2

� �e

a � d

c2

� 0

a � b2

�a2

c � e2

�e2

b � d2

�d2

�cb

�d2

, e2�.�a

2, 0�, �a � b

2,

c2�, �b � d

2,

c � e2 �,

�d, e�,�b, c�,�a, 0�,�0, 0�,

( , 0)a(0, 0)

( , )b c

( , )d e

a2

0,( )

e2

d2

,( )

b d+2

c e+2

,( )

a b+2

c2

,( )

y

x

95. Consider the figure below in which the four points arecollinear. Since the triangles are similar, the result imme-diately follows.

( , )x y11

( , )x y1*

1*

( , )x y22 ( , )x y2*

2*

y

x

y2� � y1

x2� � x1

��

y2 � y1

x2 � x1

97. True.

m2 � �1

m1

bx � ay � c2 ⇒ y �ba

x �c2

a ⇒ m2 �

ba

ax � by � c1 ⇒ y � �ab

x �c1

b ⇒ m1 � �

ab

96. If then Let be a line withslope that is perpendicular to Then Hence, and are parallel. Therefore,and are also perpendicular.L1

L2L3m2 � m3 ⇒ L2

m1m3 � �1.L1.m3

L3m1m2 � �1.m1 � �1�m2,

98. False; if is positive, then is negative.m2 � �1�m1m1

Page 22: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.3 Functions and Their Graphs

22 Chapter P Preparation for Calculus

1. (a) Domain of

Range of

Domain of

Range of

(b)

(c) for

(d) for

(e) for 1 and 2x � �1,g�x� � 0

x � 1f �x� � 2

x � �1f �x� � g�x�

g�3� � �4

f ��2� � �1

�4 ≤ y ≤ 4g:

�3 ≤ x ≤ 3g:

�3 ≤ y ≤ 5f :

�4 ≤ x ≤ 4f : 2. (a) Domain of

Range of

Domain of

Range of

(b)

(c) for and

(d) for

(e) for x � �1g�x� � 0

x � �4, 4f �x� � 2

x � 4x � �2f �x� � g�x�

g�3� � 2

f ��2� � �2

�4 ≤ y ≤ 2g:

�4 ≤ x ≤ 5g:

�4 ≤ y ≤ 4f :

�5 ≤ x ≤ 5f :

3. (a)

(b)

(c)

(d) f �x � 1� � 2�x � 1� � 3 � 2x � 5

f �b� � 2b � 3

f ��3� � 2��3� � 3 � �9

f �0� � 2�0� � 3 � �3 4. (a)

(b)

(c) undefined

(d) f �x � �x� � �x � �x � 3

f ��5� � ��5 � 3 � ��2,

f �6� � �6 � 3 � �9 � 3

f ��2� � ��2 � 3 � �1 � 1

5. (a)

(b)

(c)

(d) g�t � 1� � 3 � �t � 1�2 � �t2 � 2t � 2

g��2� � 3 � ��2�2 � 3 � 4 � �1

g��3� � 3 � ��3�2� 3 � 3 � 0

g�0� � 3 � 02 � 3 6. (a)

(b)

(c)

(d)

� �t � 4�2t � t3 � 8t2 � 16t

g�t � 4� � �t � 4�2�t � 4 � 4�

g�c� � c2�c � 4� � c3 � 4c2

g�32� � �3

2�2�32 � 4� �

94��5

2� � �458

g�4� � 42�4 � 4� � 0

9.f �x � �x� � f �x�

�x�

�x � �x�3 � x3

�x�

x3 � 3x2�x � 3x��x�2 � ��x�3 � x3

�x� 3x2 � 3x�x � ��x�2, �x � 0

11.

�1 � �x � 1

�x � 2��x � 1�

1 � �x � 1

1 � �x � 1�

2 � x

�x � 2��x � 1�1 � �x � 1��

�1�x � 1�1 � �x � 1�

, x � 2

f �x� � f �2�x � 2

��1��x � 1 � 1�

x � 2

7. (a)

(b)

(c) f ��

3� � cos�2��

3�� � cos2�

3� �

12

f ���

4� � cos�2���

4�� � cos���

2� � 0

f �0� � cos�2�0�� � cos 0 � 1 8. (a)

(b)

(c) f �2�

3 � � sin�2�

3 � ��32

f �5�

4 � � sin�5�

4 � ���2

2

f ��� � sin � � 0

12.f �x� � f �1�

x � 1�

x3 � x � 0x � 1

�x�x � 1��x � 1�

x � 1� x�x � 1�, x � 1

10.f �x� � f �1�

x � 1�

3x � 1 � �3 � 1�x � 1

�3�x � 1�

x � 1� 3, x � 1

Page 23: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.3 Functions and Their Graphs 23

13.

Domain:

Range: ���, 0�

x � 3 ≥ 0 ⇒ ��3, ��

h�x� � ��x � 3 14.

Domain:

Range: ��5, ��

���, ��

g�x� � x2 � 5 15.

Domain: all k an integer

Range: ���, �1� � �1, ��

t � 4k � 2,

�t4

��2k � 1��

2 ⇒ t � 4k � 2

f �t� � sec �t4

16.

Domain: all k an integer

Range: ���, ��

t � k�,

h�t� � cot t 17.

Domain:

Range: ���, 0� � �0, ��

���, 0� � �0, ��

f �x� �1x

18.

Domain:

Range: ���, 0� � �0, ��

���, 1� � �1, ��

g�x� �2

x � 1

19.

and

and

Domain: 0 ≤ x ≤ 1

x ≤ 1x ≥ 0

1 � x ≥ 0x ≥ 0

f �x� � �x � �1 � x 20.

Domain: or

Domain: ���, 1� � �2, ��

x ≤ 1x ≥ 2

�x � 2��x � 1� ≥ 0

x2 � 3x � 2 ≥ 0

f �x� � �x2 � 3x � 2 21.

Domain: all n an integerx � 2n�,

cos x � 1

1 � cos x � 0

g�x� �2

1 � cos x

22.

Domain: all

n integer5�

6� 2n�,

x ��

6� 2n�,

sin x �12

sin x �12

� 0

h�x� �1

sin x � �1�2� 23.

Domain: all x � �3

x � 3 � 0

x � 3 � 0

f �x� �1

x � 3 24.

Domain: all x � ±2

�x � 2��x � 2� � 0

x2 � 4 � 0

g�x� �1

x2 � 4

25.

(a)

(b)

(c)

(d)

(Note: for all t)

Domain:

Range: ���, 1� � �2, ��

���, ��

t2 � 1 ≥ 0

f �t2 � 1� � 2�t2 � 1� � 2 � 2t2 � 4

f �2� � 2�2� � 2 � 6

f �0� � 2�0� � 2 � 2

f ��1� � 2��1� � 1 � �1

f �x� � 2x � 1,

2x � 2,

x < 0

x ≥ 026.

(a)

(b)

(c)

(d)

(Note: for all s)

Domain:

Range: �2, ��

���, ��

s2 � 2 > 1

f �s2 � 2� � 2�s2 � 2�2 � 2 � 2s 4 � 8s2 � 10

f �1� � 12 � 2 � 3

f �0� � 02 � 2 � 2

f ��2� � ��2�2 � 2 � 6

f �x� � x2 � 2,

2x2 � 2,

x ≤ 1

x > 1

Page 24: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

24 Chapter P Preparation for Calculus

27.

(a)

(b)

(c)

(d)

Domain:

Range: ���, 0� � �1, ��

���, ��

f �b2 � 1� � ��b2 � 1� � 1 � �b2

f �3� � �3 � 1 � �2

f �1� � �1 � 1 � 0

f ��3� � �3 � 1 � 4

f �x� � x � 1,

�x � 1,

x < 1

x ≥ 128.

(a)

(b)

(c)

(d)

Domain:

Range: �0, ��

��4, ��

f �10� � �10 � 5�2 � 25

f �5� � �5 � 4 � 3

f �0� � �0 � 4 � 2

f ��3� � ��3 � 4 � �1 � 1

f �x� � �x � 4,

�x � 5�2,

x ≤ 5

x > 5

29.

Domain:

Range:

−2−4 2 4

2

4

6

8

x

y

���, ��

���, ��

f �x� � 4 � x 30.

Domain:

Range:

2

6

4

2 4 6

y

x

���, 0� � �0, ��

���, 0� � �0, ��

g�x� �4x

31.

Domain:

Range:

1 2 3

1

2

x

y

�0, ��

�1, ��

h�x� � �x � 1

32.

Domain:

Range:

2 4 6− 4− 6

4

8

6

y

x

���, ��

���, ��

f �x� �12 x3 � 2 33.

Domain:

Range:

−1−2−3−4 1 2 3 4

−2

−3

1

2

4

5

x

y

�0, 3�

��3, 3�

f �x� � �9 � x2 34.

Domain:

Range:

y-intercept:

x-intercept:

(0, 2)− 2, 0( (

1 2 3 4−1

−2

−3

−4

4

3

−4 −3 −2

y

x

���2, 0��0, 2�

��2, 2�2� � ��2, 2.83�

��2, 2�

f �x� � x � �4 � x2

35.

Domain:

Range: ��2, 2�

���, ��

t

y

2 3

−1

1

2

g�t� � 2 sin � t 36.

Domain:

Range: ��5, 5�

���, ��

ππ

5

− 5

θ−2 2

4

321

yh�� � �5 cos

2

Page 25: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.3 Functions and Their Graphs 25

37. The student travels during the first

4 minutes. The student is stationary for the following

2 minutes. Finally, the student travels

during the final 4 minutes.

6 � 210 � 6

� 1 mi�min

2 � 04 � 0

�12

mi�min 38.

t

27

t t t1 2 3

9

18

d

39.

y is not a function of x. Some vertical lines intersectthe graph twice.

x � y 2 � 0 ⇒ y � ±�x 40.

y is a function of x. Vertical lines intersect the graphat most once.

�x2 � 4 � y � 0 ⇒ y � �x2 � 4

41. y is a function of x. Vertical lines intersect the graphat most once.

42.

y is not a function of x. Some vertical lines intersectthe graph twice.

y � ±�4 � x2

x2 � y2 � 4

43.

y is not a function of x since there are two values of y forsome x.

x2 � y2 � 4 ⇒ y � ±�4 � x2 44.

y is a function of x since there is one value of y foreach x.

x2 � y � 4 ⇒ y � 4 � x2

45.

y is not a function of x since there are two values of y forsome x.

y2 � x2 � 1 ⇒ y � ±�x2 � 1 46.

y is a function of x since there is one value of y foreach x.

x2y � x2 � 4y � 0 ⇒ y �x2

x2 � 4

47. is a horizontal shift 5 units to the left. Matches d.y � f �x � 5� 48. is a vertical shift

5 units downward. Matches b.y � f �x� � 5 49. is a reflection in

the axis, a reflection in the axis,and a vertical shift downward 2 units. Matches c.

x-y-y � �f ��x� � 2

50. is a horizontalshift 4 units to the right, followedby a reflection in the axis.Matches a.

x-

y � �f �x � 4� 51. is a horizontalshift to the left 6 units, and a vertical shift upward 2 units.Matches e.

y � f �x � 6� � 2 52. is a horizontalshift to the right 1 unit, and a vertical shift upward 3 units.Matches g.

y � f �x � 1� � 3

53. (a) The graph is shifted 3 units to the left.

—CONTINUED—

x

−6

−2

−4

4

−4−6 −2 2 4

y

(b) The graph is shifted 1 unit to the right.

x

−6

−2

−4

4

2

−2 2 4 6 8

y

(c) The graph is shifted 2 unitsupward.

x

−2

4

6

2

−4 −2 2 4 6

y

Page 26: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

26 Chapter P Preparation for Calculus

53. —CONTINUED—

(d) The graph is shifted 4 unitsdownward.

x

−2

−8

−6

−4

−4 −2 2 4 6

y

(e) The graph is stretched vertically by a factor of 3.

x

−2

−8

−10

−6

−4

−4 −2 4 6

y

(f ) The graph is stretchedvertically by a factor of

x

−6

4

2

−4 −2 2 4 6

y

14.

54. (a)

Shift f right 4 units

1 2 3 5 6 7−1

1

3

2

4

−2

(0, 3)−

(6, 1)

−4

y

x

g�0� � f ��4� � �3

g�6� � f �2� � 1

g�x� � f �x � 4� (b)

Shift f left 2 units

1−1

3

2

4

−2

−4

−4 −3

−3

−5−7 −6

(0, 1)

( 6, 3)− −

y

x

g�x� � f �x � 2� (c)

Vertical shift upwards4 units

1 2 3−1−3−5 −4 −2

−2

1

4

6

5

2

( 4, 1)−

(2, 5)

y

x

g�x� � f �x� � 4

(d)

Vertical shift down 1 unit

2 3−1−3

−3

−5

−5

−6

−4

−4

−2

1

2

(2, 0)

( 4, 4)− −

y

x

g�x� � f �x� � 1 (e)

1 2 3−1−3

−3

−5

−5

−6

−4

−4

−2

1

2(2, 2)

( 4, )− −6

y

x

g��4� � 2f ��4� � �6

g�2� � 2 f �2� � 2

g�x� � 2f �x� (f )

1 2 3−1

−3

−3

−5

−5

−6

−4

−4

−2

1

2

3

1

2

2

− −4,

2,

( )

( )

y

x

g��4� �12 f ��4� � �

32

g�2� �12 f �2� �

12

g�x� �12 f �x�

55. (a)

Vertical shift 2 units upward

4

3

2

1

4321

y

x

y � �x � 2 (b)

Reflection about the x-axis

4321

−1

−2

1

−3

x

y

y � ��x (c)

Horizontal shift 2 units to the right

643

3

2

1

1

−1

−2

2

4

5x

y

y � �x � 2

56. (a) is a horizontal shift units to the left, followed by a vertical shift 1 unit upwards.

(b) is a horizontal shift 1 unit to the right followed by a reflection about the x-axis.h�x� � �sin�x � 1�

��2h�x� � sin�x � ���2�� � 1

Page 27: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.3 Functions and Their Graphs 27

58.

(a)

(b)

(c)

(d)

(e)

(f ) g� f �x�� � g�sin x� � � sin x

f �g�x�� � f ��x� � sin��x�

g� f ���4�� � g�sin���4�� � g��2�2� � � ��2�2� ���2

2

g� f �0�� � g�0� � 0

f �g�1�2�� � f ���2� � sin���2� � 1

f �g�2�� � f �2�� � sin�2�� � 0

g�x� � �xf �x� � sin x,

57. (a)

(b)

(c) g� f �0�� � g�0� � �1

g� f �1�� � g�1� � 0

f �g�1�� � f �0� � 0 (d)

(e)

(f ) g� f �x�� � g��x � � ��x �2� 1 � x � 1, �x ≥ 0�

f �g�x�� � f �x2 � 1� � �x2 � 1

f �g��4�� � f �15� � �15

59.

Domain:

Domain:

No. Their domains are different. � f g� � �g f � for x ≥ 0.

���, ��

�g f ��x� � g� f �x�� � g�x2� � �x2 � x�0, ��

� f g��x� � f �g�x�� � f ��x� � ��x�2� x, x ≥ 0

f �x� � x2, g�x� � �x 60.

Domain:

Domain:

No, f g � g f.

���, ��

�g f ��x� � g�x 2 � 1� � cos�x 2 � 1�

���, ��

� f g��x� � f �g�x�� � f �cos x� � cos2 x � 1

f �x� � x2 � 1, g�x� � cos x

61.

Domain: all

Domain: all

No, f g � g f.

x � 0

�g f ��x� � g� f �x�� � g �3x� � �3

x�2

� 1� 9x2 � 1 �

9 � x2

x2

x � ±1

� f g��x� � f �g�x�� � f �x2 � 1� �3

x2 � 1

f �x� �3x, g�x� � x2 � 1

62.

Domain:

You can find the domain of by determining the intervals where and are both positive, or both negative.

Domain: ���, �12�, �0, ��

0 1 2−1 12

−2 −x

+ + + ++++ − −

x�1 � 2x�g f

�g f ��x� � g�1x� ��1

x� 2 ��1 � 2x

x

��2, ��

� f g��x� � f ��x � 2� �1

�x � 2

63. (a)

(b)

(c) which is undefinedg� f �5�� � g��5�,

g� f �2�� � g�1� � �2

� f g��3� � f �g�3�� � f ��1� � 4 (d)

(e)

(f ) which is undefinedf �g��1� � f ��4�,

�g f ���1� � g� f ��1�� � g�4� � 2

� f g���3� � f �g��3�� � f ��2� � 3

Page 28: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

28 Chapter P Preparation for Calculus

64.

represents the area of the circle at time t.�A r��t�

�A r��t� � A�r�t�� � A�0.6t� � ��0.6t�2 � 0.36�t 2

65.

Let and

Then,

[Other answers possible]

� f g h��x� � f �g�x � 1�� � f �2�x � 1�� � �2�x � 1� � �2x � 2 � F�x�.

f �x� � �x.g�x� � 2xh�x� � x � 1,

F�x� � �2x � 2

66.

Let and

Then,

[Other answers possible]

� f g h��x� � f �g�1 � x�� � f �sin�1 � x�� � �4 sin�1 � x� � F�x�.

h�x� � 1 � x.g�x� � sin xf �x� � �4x,

F�x� � �4 sin�1 � x�

67.

Even

f ��x� � ��x�2�4 � ��x�2� � x2�4 � x2� � f �x� 68.

Odd

f ��x� � 3��x � � 3�x � �f �x�

69.

Odd

f ��x� � ��x� cos��x� � �x cos x � �f �x�

70.

Even

f ��x� � sin2��x� � sin��x� sin��x� � ��sin x���sin x� � sin2 x

71. (a) If f is even, then is on the graph.

(b) If f is odd, then is on the graph.�32 , �4��3

2 , 4� 72. (a) If f is even, then is on the graph.

(b) If f is odd, then is on the graph.��4, �9�

��4, 9�

73. f is even because the graph is symmetric about the axis.

is neither even nor odd.

is odd because the graph is symmetric about the origin.h

g

y-

74. (a) If f is even, then the graph is symmetric about the axis.

x

y

f

2

2

−2

−4

−6

4

6

4 6−6 −4 −2

y- (b) If is odd, then the graph is symmetric about the origin.

x

y

f

2

2

−2

−4

−6

4

6

4 6−6 −4 −2

f

75.

�4 ≤ x ≤ 0 f �x� � �2x � 5,

y � �2x � 5

y � 5 � �2�x � 0�x

y

−4−6 2 4 6

−6

2

4

6

(−4, 3)

(0, −5)

Slope �3 � 5

�4 � 0� �2

Page 29: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.3 Functions and Their Graphs 29

76.

1 ≤ x ≤ 5 f �x� �34

x �54

,

y �34

x �54

y � 2 �34

�x � 1�

x

y

(1, 2)

(5, 5)

−1−2 1 2 3 4 5−1

−2

1

2

3

4

5 Slope �

5 � 25 � 1

�34

77.

x ≤ 0 f �x� � ���x,

y � ���x

y2 � �x

x

y

−1−2−3−4−5 1

−2

−3

1

2

3 x � y2 � 0

78.

�2 ≤ x ≤ 2 f �x� � ��4 � x2,

y � ��4 � x2

y2 � 4 � x2

x

y

−1 1

−1

1

x2 � y2 � 4

79. Matches (ii). The function is Since satisfies the equation, Thus, g�x� � �2x2.c � �2.

�1, �2�g�x� � cx2. 80. Matches (i). The function is Since satisfies the equation, Thus, f �x� � �1�4�x.c � 1�4.

�1, 1�4�f �x� � cx.

81. Matches (iv). The function is since it must beundefined at Since satisfies the equation,

Thus, r�x� � 32�x.c � 32.�1, 32�x � 0.

r�x� � c�x, 82. Matches (iii). The function is Since satisfies the equation, Thus, h�x� � 3�x.c � 3.

�1, 3�h�x� � c�x.

83. (a)

(b) If then the program would turn on (and off) one hour later.

(c) If then the overall temperature would be reduced 1 degree.H�t� � T�t� � 1,

H�t� � T�t � 1�,

T�4� � 16, T�15� � 23

84. (a) For each time t, there corresponds a depth d.

(b) Domain:

Range:

(c)

t

d

5

10

15

20

25

30

1 2 3 4 5 6

0 ≤ d ≤ 30

0 ≤ t ≤ 5

85. (a)

(b) A�15� � 345 acres�farm

10

100

200

300

400

500

20 30 40 50t

A

86. (a)

(b)

� 0.00078125x 2 � 0.003125x � 0.029

H� x1.6� � 0.002� x

1.6�2

� 0.005� x1.6� � 0.029

0 1000

25

Page 30: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

30 Chapter P Preparation for Calculus

87.

If then

If then

If then

Thus,

f �x� � 2�1 � x�,2, 2�x � 1�,

x < 00 ≤ x < 2 .x ≥ 2

f �x� � x � �x � 2� � 2x � 2 � 2�x � 1�.x ≥ 2,

f �x� � x � �x � 2� � 2.0 ≤ x < 2,

f �x� � �x � �x � 2� � �2x � 2 � 2�1 � x�.x < 0,

f �x� � x � x � 2

88. has one zero. has three zeros. Every cubic polynomial has at least one zero. Given we have

as and as if Furthermore, as and as if Since the graph has no breaks, the

graph must cross the x-axis at least one time.A < 0.x →�p → ��x → ��

p →�A > 0.x →�p →�x → ��p → ��p�x� � Ax3 � Bx2 � Cx � D,

− 3 3

−2

p

p

2

1

2p2�x� � x3 � xp1�x� � x3 � x � 1

89.

Odd

� �f �x�

� ��a2n�1x2n�1 � . . . � a3 x3 � a1x�

f ��x� � a2n�1��x�2n�1 � . . . � a3��x�3 � a1��x�

90.

Even

� f �x�

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

f ��x� � a2n��x�2n � a2n�2��x�2n�2 � . . . � a2��x�2 � a0

91. Let where f and g are even. Then

.

Thus, is even. Let where f and g are odd. Then

Thus, is even.F�x�

F��x� � f ��x�g��x� � ��f �x����g�x�� � f �x�g�x� � F�x�.

F�x� � f �x�g�x�F�x�

F��x� � f ��x�g��x� � f �x�g�x� � F�x�

F�x� � f �x�g�x�

92. Let where f is even and g is odd. Then

Thus, is odd.F�x�

F��x� � f ��x�g��x� � f �x���g�x�� � �f �x�g�x� � �F�x�.

F�x� � f �x�g�x�

93. (a) (c)

Domain:

(b)

The dimensions for maximum volume are cm.

The dimensions for maximum volume appear to be cm.4 � 16 � 16

4 � 16 � 16

12

−100

−1

1100

0 < x < 12

V � x�24 � 2x�2 � 4x�12 � x2�x length and width volume

1

2

3

4

5

6 6�24 � 2�6��2 � 86424 � 2�6�

5�24 � 2�5��2 � 98024 � 2�5�

4�24 � 2�4��2 � 102424 � 2�4�

3�24 � 2�3��2 � 97224 � 2�3�

2�24 � 2�2��2 � 80024 � 2�2�

1�24 � 2�1��2 � 48424 � 2�1�

Page 31: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.4 Fitting Models to Data 31

98. False; let Then and Thus, 3f �x� � f �3x�.3f �x� � 3x2.f �3x� � �3x�2 � 9x2f �x� � x2.

94.

,

.L � �x2 � y2 ��x2 � � 2xx � 3�

2

y �6

x � 3� 2 �

2xx � 3

y � 2 �6

x � 3

By equating slopes, y � 20 � 3

�0 � 2x � 3

95. False; let

Then but .�3 � 3f ��3� � f �3� � 9,

f �x� � x2.

96. True 97. True, the function is even.

99. First consider the portion of in the first quadrant:and shown below.

The area of this region is 1 �12 �

32.

x

y

−1 2

−1

1

2

(2, 1)(0, 1)

(0, 0) (1, 0)

x � y ≤ 1;0 ≤ y ≤ 1x ≥ 0,R By symmetry, you obtain the entire region

The area of is

[49th competition, Problem A1, 1988]

4�32� � 6.R

x

y

−2 1 2

−2

2

(2, 1)

(2, −1)

(−2, 1)

(−2, −1)

R:

100. Let be a constant polynomial.

Then and

Thus, Since this is true for all real numbers c, f is the identity function: f �x� � x.f �c� � c.

g� f �x�� � c.f �g�x�� � f �c�

g�x� � c

Section P.4 Fitting Models to Data

1. Quadratic function 2. Trigonometric function 3. Linear function 4. No relationship

5. (a), (b)

Yes. The cancer mortality increases linearly withincreased exposure to the carcinogenic substance.

(c) If x � 3, then y � 136.

x

y

3 6 9 12 15

50

100

150

200

250

6. (a)

No, the relationship does not appear to be linear.

(b) Quiz scores are dependent on several variables such asstudy time, class attendance, etc. These variables maychange from one quiz to the next.

0 200

20

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32 Chapter P Preparation for Calculus

7. (a)

(b)

The model fits well.

(c) If then d � 0.066�55� � 3.63 cm.F � 55,

00

10

F d= 15.13 + 0.1

125

d � 0.066F or F � 15.1d � 0.1 8. (a)

(b)

The model fits well.

(c) If t � 2.5, s � 24.65 meters�second.

45

5

−5

−1

s � 9.7t � 0.4

9. (a) Using a graphing utility,

correlation coefficient

(b)

(c) The data indicates that greater per capita electricityconsumption tends to correspond to greater per capitagross national product.

The data for Hong Kong, Venezuela and South Koreadiffer most from the linear model.

(d) Removing the data andyou obtain the model

with r � 0.968.y � 0.134x � 0.28�167, 17.3�,

�113, 5.74��118, 25.59�,

30

00

180

y = 0.124x + 0.82

r � 0.838

y � 0.124x � 0.82. 10. (a) Linear model:

(b)

The fit is very good.

(c) When

H � �0.3323�500� � 612.9333 � 446.78.

t � 500,

00 1300

600

H � �0.3323t � 612.9333

11. (a)

(b)

For y1 � y2 � y3 � 31.06 cents�mile.t � 12,

80

0

y1

y2

y3

y1 + y2 + y3

15

y3 � 0.0917t � 0.7917

y2 � 0.1095t � 2.0667

y1 � 0.0343t3 � 0.3451t2 � 0.8837t � 5.6061 12. (a)

(b)

(c) When .x � 2, S � 583.98 pounds

00

14

25000

S � 180.89x 2 � 205.79x � 272

13. (a) Linear:

Cubic:

(b)

(c) The cubic model is better.

90

1325

0

y1

y2

y2 � �0.1289t3 � 2.235t2 � 4.86t � 35.2

y1 � 4.83t � 28.6 (d)

(e) For Linear model million

Cubic model million

(f ) Answers will vary.

y2 � 51.5

y1 � 96.2t � 14:

90

1325

0

y � �0.084t2 � 5.84t � 26.7

Page 33: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Section P.4 Fitting Models to Data 33

14. (a)

(b)

(c) The curve levels off for

(d)

(e) The model is better for low speeds.

0 1000

21

t � 0.002s2 � 0.0346s � 0.183

s < 20.

20 1000

21

t � 0.00271s2 � 0.0529s � 2.671 15. (a)

(b)

(c) If horsepower.x � 4.5, y � 214

00

7

300

y � �1.806x3 � 14.58x2 � 16.4x � 10

16. (a)

(b)

(c) For pounds per square inch.

(d) The model is based on data up to 100 pounds persquare inch.

T � 300�F, p � 68.29

0 110150

350

T � 2.9856 � 10�4 p3 � 0.0641 p2 � 5.2826p � 143.1 17. (a) Yes, y is a function of t. At each time t, there is oneand only one displacement y.

(b) The amplitude is approximately

The period is approximately

(c) One model is .

(d)

00

4

0.9

(0.125, 2.35)

(0.375, 1.65)

y � 0.35 sin�4� t� � 2

2�0.375 � 0.125� � 0.5.

�2.35 � 1.65��2 � 0.35.

18. (a)

One model is

(b)

00

13

100

C�t� � 58 � 27 sin�� t6

� 4.1�.

H�t� � 84.4 � 4.28 sin�� t6

� 3.86� (c)

(d) The average in Honolulu is 84.4.

The average in Chicago is 58.

(e) The period is 12 months (1 year).

(f ) Chicago has greater variability �27 > 4.28�.

00

13

100

19. Answers will vary. 20. Answers will vary.

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34 Chapter P Preparation for Calculus

Review Exercises for Chapter P

1.

y-intercept

x-intercepty � 0 ⇒ 0 � 2x � 3 ⇒ x �32 ⇒ �3

2 , 0�x � 0 ⇒ y � 2�0� � 3 � �3 ⇒ �0, �3�

y � 2x � 3

2.

y-intercept

x-interceptsy � 0 ⇒ 0 � �x � 1��x � 3� ⇒ x � 1, 3 ⇒ �1, 0�, �3, 0�

x � 0 ⇒ y � �0 � 1��0 � 3� � 3 ⇒ �0, 3�

y � �x � 1��x � 3�

3.

y-intercept

x-intercepty � 0 ⇒ 0 �x � 1x � 2

⇒ x � 1 ⇒ �1, 0�

x � 0 ⇒ y �0 � 10 � 2

�12

⇒ �0, 12�

y �x � 1x � 2

4.

and are both impossible. No intercepts.y � 0x � 0

xy � 4

5. Symmetric with respect to y-axis since

x2y � x2 � 4y � 0.

��x�2y � ��x�2 � 4y � 0

6. Symmetric with respect to y-axis since

y � x 4 � x2 � 3.

y � ��x�4 � ��x�2 � 3

7.

−1 1 2 3

−1

−2

1

2

3

x

y

y � �12 x �

32 8.

Slope: 2

y-intercept:

2 3−1−2

−1

−2

−3

1

y

x

�0, �3�

y � 2x � 3

4x � 2y � 6 9.

Slope:

y-intercept:

−1−2−3 1

−1

2

3

x

y

65

25

y �25 x �

65

�25 x � y �

65

�13 x �

56 y � 1

10.

Slope:

y-intercept: �0, 53��

215

y � �2

15 x �53

2x � 15y � 25

− 4 4 8 12

−1

−2

1

3

y

x

0.02x � 0.15y � 0.25 11.

x

y

−5−10 5

5

10

y � 7 � 6x � x2

Page 35: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Review Exercises for Chapter P 35

12.

2 4 8

2

4

6

8

10

−2

y

x

y � x�6 � x� 13.

Domain:

1 2 3 4 5

1

2

3

4

5

x

y

���, 5�

y � �5 � x 14.

−1 2 3 4 5 6−1

−2

−3

−4

−5

−6

y

x

y � �x � 4� � 4

15.

Xmin = -5Xmax = 5Xscl = 1Ymin = -30Ymax = 10Yscl = 5

y � 4x2 � 25 16.

Xmin = -40Xmax = 40Xscl = 10Ymin = -40Ymax = 40Yscl = 10

y � 8 3�x � 6 17.

Point: �4, 1�

y � 1

x � 4

7x � 28

4x � 4y � 20

3x � 4y � 8

19. You need factors and Multiply by x to obtain origin symmetry.

� x3 � 4x

y � x�x � 2��x � 2�

�x � 2�.�x � 2� 20.

(a) and

(b) and

(c)

(d) �1 � k��1�3 ⇒ k � 1 ⇒ y � x3

0 � k�0�3 ⇒ any k will do!

y � �18 x31 � k��2�3 ⇒ k � �

18

y � 4x34 � k�1�3 ⇒ k � 4

y � kx3

18.

No real solutionNo points of intersectionThe graphs of and do not intersect.y � x2 � 7y � x � 1

0 � x2 � x � 6

�x � 1� � x2 � 7

y � x � 1

−12 12

14

−2

21.

Slope ��5�2� � 15 � �3�2� �

3�27�2

�37

x

y

1 2 3 4 5

1

2

3

4

5

, 132

52

5,( )

( )

22.

The line is vertical and hasno slope.

2

6

4

8

12

10

14

−21 2 3 4 5 6

(7, 12)

(7, 1)−

y

x

23.

t �73

1 � t � �43

1 � t1 � 0

�1 � 5

1 � ��2�

Page 36: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

36 Chapter P Preparation for Calculus

24.

t � �533

�53 � 3t

�44 � 9 � 3t

4

�3 � t�

�3�11

3 � ��1��3 � t

�3 � 6

�3 � 825.

y

x

(0, −5)

−4

−4

−8

−2−2

2

2

4

4 6 8

2y � 3x � 10 � 0

y �32x � 5

y � ��5� �32�x � 0� 26.

Horizontal line

y

x−6 −4

−4

−2

2

4

8

−2 2 4 6

(−2, 6)

y � 6

y � 6 � 0�x � ��2��

27.

y

x−8

−8

−6

−4

−2

2

4

−6 −4 2 4

(−3, 0)

3y � 2x � 6 � 0

y � �23x � 2

y � 0 � �23�x � ��3�� 28. m is undefined. Line is vertical.

y

x−4

−4

−6

−2−2

2

4

6

2 4 6 8

(5, 4)

x � 5

29. (a)

(b) Slope of line is

(c)

(d)

x � 2 � 0

x � �2

2x � y � 0

y � �2x

m �4 � 0

�2 � 0� �2

0 � 5x � 3y � 22

3y � 12 � 5x � 10

y � 4 �53

�x � 2�

53

.

0 � 7x � 16y � 78

16y � 64 � 7x � 14

y � 4 �7

16�x � 2� 30. (a)

(b) Slope of perpendicular line is 1.

(c)

(d)

y � 3 � 0

y � 3

0 � x � y � 2

y � x � 2

y � 3 � 1�x � 1�

m �4 � 32 � 1

� 1

0 � x � y � 2

y � x � 2

y � 3 � 1�x � 1�

2x � 3y � 11 � 0

3y � 9 � �2x � 2

y � 3 � �23

�x � 1�

31. The slope is

V�3� � �850�3� � 12,500 � $9950

V � �850t � 12,500.�850.

Page 37: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Review Exercises for Chapter P 37

34.

Function of x since there is onevalue for y for each x.

1 2 3−1−2−3

1

3

2

4

6

5

y

x

x2 � y � 0 35.

Function of x since there is onevalue of y for each x.

x

y

−1−2 3 4

−2

3

4

y � x2 � 2x 36.

Not a function of x since there are two values of yfor some x.

3 6 12−3−9 −6−12−1

−2

−4

1

4

2

y

x

x � 9 � y2

37.

(a) does not exist.

(b)

��1

1 � �x, �x � �1, 0

f �1 � �x� � f �1��x

11 � �x

�11

�x�

1 � 1 � �x�1 � �x��x

f �0�

f �x� �1x

38. (a) (because )

(b)

(c) f �1� � �1 � 2� � 1

f �0� � �0 � 2� � 2

�4 < 0f ��4� � ��4�2 � 2 � 18

39. (a) Domain:

Range:

(b) Domain: all

Range: all

(c) Domain: all x or

Range: all y or ���, ��

���, ��

y � 0 or ���, 0� � �0, ��

x � 5 or ���, 5� � �5, ��

0, 6�

36 � x2 ≥ 0 ⇒ �6 ≤ x ≤ 6 or �6, 6� 40. and

(a)

(b)

(c) g� f �x�� � g�1 � x2� � 2�1 � x2� � 1 � 3 � 2x2

f �x�g�x� � �1 � x2��2x � 1� � �2x3 � x2 � 2x � 1

f �x� � g�x� � �1 � x2� � �2x � 1� � �x2 � 2x

g�x� � 2x � 1f �x� � 1 � x2

32. (a)

(b)

(c)

t 5034.48 hours to break even

7.25t � 36,500

30t � 22.75t � 36,500

R � 30t

� 22.75t � 36,500

C � 9.25t � 13.50t � 36,500 33.

Not a function of x since there are two values of y forsome x.

x

y

1 2 3 4 5 6−1

−2

−3

1

2

3

y � ±�x

x � y2 � 0

Page 38: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

38 Chapter P Preparation for Calculus

42.

(a) The graph of g is obtained from f by a vertical shiftdown 1 unit, followed by a reflection in the x-axis:

� �x3 � 3x2 � 1

g�x� � � f �x� � 1�

−6 6

−6

(0, 0)

(2, 4)−

6

f �x� � x3 � 3x2

(b) The graph of g is obtained from f by a vertical shiftupwards of 1 and a horizontal shift of 2 to the right.

� �x � 2�3 � 3�x � 2�2 � 1

g�x� � f �x � 2� � 1

43. (a) Odd powers:

The graphs of f, g, and h all rise to the right and fall tothe left. As the degree increases, the graph rises andfalls more steeply. All three graphs pass through thepoints �0, 0�, �1, 1�, and ��1, �1�.

3

−2

−3

f

h

g2

f �x� � x, g�x� � x3, h�x� � x5 Even powers:

The graphs of f, g, and h all rise to the left and to theright. As the degree increases, the graph rises moresteeply. All three graphs pass through the points �1, 1�, and ��1, 1�.

�0, 0�,

30

−3

h

g

f

4

f �x� � x2, g�x� � x4, h�x� � x6

(b) will look like but rise and fall even more steeply.

will look like but rise even more steeply.h�x� � x6,y � x8

h�x� � x5,y � x7

41. (a)

(c)

42

1

1

2

2

x

y

2c

0c

c 2

f �x� � �x � 2�3 � c, c � �2, 0, 2

3

3

1

223x

y

2

c

c

0

c 2

f �x� � x3 � c, c � �2, 0, 2 (b)

(d)

3

3

2

1

21123x

y

0

2

c

c

2c

f �x� � cx3, c � �2, 0, 2

3

1

22

2

3

x

y0c

2c

2c

f �x� � �x � c�3, c � �2, 0, 2

44. (a)

−4 10

−25

100

f �x� � x2�x � 6�2 (b)

−2 10

−100

300

g�x� � x3�x � 6�2 (c)

−4 10

−800

200

h�x� � x3�x � 6�3

Page 39: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Problem Solving for Chapter P 39

45. (a)

A � xy � x�12 � x� � 12x � x2

y � 12 � x

2x � 2y � 24

x x

y

y (b) Domain:

(c) Maximum area is In general, the maximumarea is attained when the rectangle is a square. In thiscase, x � 6.

A � 36.

00

12

40

0 < x < 12

49. (a) Yes, y is a function of t. At each time t, there is oneand only one displacement y.

(b) The amplitude is approximately

.

The period is approximately 1.1.

(c) One model is

(d)

0 2.2

−0.5

0.5

(0.5, −0.25)

(1.1, 0.25)

y �14

cos�2�

1.1t� �

14

cos�5.7t�

�0.25 � ��0.25���2 � 0.25

46. For company (a) the profit rose rapidly for the first year, and then leveled off. For the second company (b), the profit dropped,and then rose again later.

48. (a)

(b)

(c) The data point is probably an error. Without this point, the new model is

y � �1.4344x � 66.4387.

�27, 44�

3300

70

y � �1.204x � 64.2667

47. (a) 3 (cubic), negative leading coefficient

(b) 4 (quartic), positive leading coefficient

(c) 2 (quadratic), negative leading coefficient

(d) 5, positive leading coefficient

Problem Solving for Chapter P

1. (a)

Center: Radius: 5

(c) Slope of line from to is

Slope of tangent line is Hence,

Tangent liney � 0 �34

�x � 6� ⇒ y �34

x �92

34

.

4 � 03 � 6

� �43

.�3, 4��6, 0�

�3, 4�

�x � 3�2 � � y � 4�2 � 25

�x2 � 6x � 9� � �y2 � 8y � 16� � 9 � 16

x2 � 6x � y2 � 8y � 0 (b) Slope of line from to is Slope of tangent line

is Hence,

Tangent line

(d)

Intersection: �3, �94�

x � 3

32

x �92

�34

x �34

x �92

y � 0 � �34

�x � 0� ⇒ y � �34

x

�34

.

43

.�3, 4��0, 0�

Page 40: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

40 Chapter P Preparation for Calculus

2. Let be a tangent line to the circle from the point Then

Setting the discriminant equal to zero,

Tangent lines: and y � ��3x � 1.y � �3x � 1

m � ±�3

4m2 � 12

16m2 � 12m2 � 12

16m2 � 4�m2 � 1��3� � 0

b2 � 4ac

�m2 � 1�x2 � 4mx � 3 � 0

x2 � �mx � 1 � 1�2 � 1

x2 � �y � 1�2 � 1

�0, 1�.y � mx � 1

3.

(a)

(c)

x1

2

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

y�H�x�

x1

2

1

3

4

−2 −1−1

−3

−4

−3−4 2 3 4

yH�x� � 2

x1

2

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

yH�x� � �1,

0,

x ≥ 0

x < 0

(b)

(d)

x1

2

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

yH��x�

x1

2

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

yH�x � 2�

(e)

x1

2

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

y12H�x� (f )

x1

1

3

4

−2 −1−1

−2

−3

−4

−3−4 2 3 4

y�H�x � 2� � 2

Page 41: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Problem Solving for Chapter P 41

4. (a)

(c)

(e)

(g) y

x−4

−4

−2

−2

2

2

4

4

f �x�

y

x−4

−4

−2

−2

2

2

4

4

�f �x�

y

x−4 −2

−4

−2

42

4

2 f �x�

y

x

−4

−3 −1

−2

1 3

4

f �x � 1� (b)

(d)

(f ) y

x−4

−4

−2

−2

2

2

4

4

f �x�

y

x−4

−4

−2

−2

2

2

4

4

f ��x�

y

x−4

−4

−2

4

4

f �x� � 1

5. (a)

Domain:

(b)

Maximum of at x � 50 m, y � 25 m.1250 m2

1100

0

1600

0 < x < 100

A�x� � xy � x�100 � x2 � � �

x2

2� 50x

x � 2y � 100 ⇒ y �100 � x

2(c)

is the maximum.x � 50 m, y � 25 mA�50� � 1250 m2

� �12

�x � 50�2 � 1250

� �12

�x2 � 100x � 2500� � 1250

A�x� � �12

�x2 � 100x�

Page 42: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

42 Chapter P Preparation for Calculus

7. The length of the trip in the water is and thelength of the trip over land is Hence,the total time is

T ��4 � x2

2�

�1 � �3 � x�2

4 hours.

�1 � �3 � x�2.�22 � x2,

6. (a)

Domain:

(b)

Maximum of at x � 50 ft, y � 37.5 ft.3750 ft2

25 50 75 100

500

1000

1500

2000

2500

3000

3500

4000

y

x

0 < x < 100

A�x� � x�2y� � x�300 � 3x2 � �

�3x2 � 300x2

4y � 3x � 300 ⇒ y �300 � 3x

4(c)

square feet is the maximum area,where x � 50 ft and y � 37.5 ft.A�50� � 3750

� �32

�x � 50�2 � 3750

� �32

�x2 � 100x � 2500� � 3750

A�x� � �32

�x2 � 100x�

8. Let d be the distance from the starting point to the beach.

� 80 km�hr

�2

1120

�1

60

�2d

d120

�d

60

Average speed �distance

time

9. (a) Slope of tangent line is less than 5.

(b) Slope of tangent line is greater than 3.

(c) Slope of tangent line is less than 4.1.

(d)

(e) Letting h get closer and closer to 0, the slope approaches 4. Hence, the slope at is 4.�2, 4�

� 4 � h, h � 0

�4h � h2

h

��2 � h�2 � 4

h

Slope �f �2 � h� � f �2�

�2 � h� � 2

Slope �4.41 � 42.1 � 2

� 4.1.

Slope �4 � 12 � 1

� 3.

Slope �9 � 43 � 2

� 5.

Page 43: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

Problem Solving for Chapter P 43

10.

(a) Slope of tangent line is greater than

(b) Slope of tangent line is less than

(c) Slope of tangent line is greater than

(d)

(e)

As h gets closer to 0, the slope gets closer to The slope is at the point �4, 2�.14

14

.

�1

�4 � h � 2, h � 0

��4 � h� � 4

h��4 � h � 2�

�4 � h � 2

h�

�4 � h � 2h

��4 � h � 2

�4 � h � 2

��4 � h � 2

h

Slope �f �4 � h� � f �4�

�4 � h� � 4

1041

.Slope �2.1 � 2

4.41 � 4�

1041

.

13

.Slope �2 � 14 � 1

�13

.

15

.Slope �3 � 29 � 4

�15

.

y

x1

1

−1

2

3

4

2 3 4 5

(4, 2)

11. (a)

(b)

Circle of radius and center ��3, 0�.�18

�x � 3�2 � y2 � 18

x2 � y2 � 6x � 9 � 0

x2 � 6x � 9 � y2 � 2x2 � 2y2

�x � 3�2 � y2 � 2�x2 � y2�

−2−4−8 2 4−2

−6

2

6

8

x

y I

x2 � y2 �2I

�x � 3�2 � y2

� �3 ± �18 � 1.2426, �7.2426 x ��6 ± �36 � 36

2

x2 � 6x � 9 � 0

x2 � 6x � 9 � 2x2

0 1 2

x

3

I

x2 �2I

�x � 3�2

Page 44: CHAPTER P Preparation for Calculuscrunchymath.weebly.com/uploads/8/2/4/0/8240213/solutions.pdf8 Chapter P Preparation for Calculus 62. The corresponding y-value is Point of intersection:

44 Chapter P Preparation for Calculus

12. (a)

If then is a vertical line. Assume

Circle

(c) As becomes very large, and

The center of the circle gets closer to and its radius approaches 0.�0, 0�,

16k�k � 1�2 → 0.

4k � 1

→ 0k

�x �4

k � 1�2

� y2 �16k

�k � 1�2,

x2 �8x

k � 1�

16�k � 1�2 � y2 �

16k � 1

�16

�k � 1�2

x2 �8x

k � 1� y2 �

16k � 1

k � 1.x � 2k � 1,

�k � 1�x2 � 8x � �k � 1�y2 � 16

�x � 4�2 � y2�k�x2 � y2�

Ix2 � y2 �

kI�x � 4�2 � y2

13.

Let Then or

Thus, and are on the curve.���2, 0��0, 0�, ��2, 0�x2 � 2.x 4 � 2x2 ⇒ x � 0y � 0.

�x2 � y2�2 � 2�x2 � y2�

�x4 � 2x2y2 � y4� � 2x2 � 2y2 � 0

x 4 � 2x2 � 1 � 2x2y2 � 2y2 � y4 � 1

�x2 � 1�2 � y22x2 � 2� � y4 � 1

�x � 1�2�x � 1�2 � y2�x � 1�2 � �x � 1�2� � y4 � 1

�x � 1�2 � y2��x � 1�2 � y2� � 1

(− 2 , 0) ( 2 , 0)

(0, 0)

x

y

−2

−2

−1

1

2

2

d1d2 � 1

14.

(a) Domain: all

Range: all

(b)

Domain: all

(c)

Domain: all

(d) The graph is not a line. It has holes at and y

x21−2

−2

1

2

�1, 1�.�0, 0�

x � 0, 1

f � f � f �x��� � f �x � 1x � �

1

1 � �x � 1x �

�11x

� x

x � 0, 1

f � f �x�� � f � 11 � x� �

1

1 � � 11 � x�

�1

1 � x � 11 � x

�1 � x

�x�

x � 1x

y � 0

x � 1

f �x� � y �1

1 � x

(b) If

x

y

−2−4−6 2 4

−2

−4

2

4

6

�x � 2�2 � y2 � 12k � 3,