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C H A P T E R 1Functions and Their Graphs
Section 1.1 Lines in the Plane . . . . . . . . . . . . . . . . . . . . . . . 2
Section 1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Section 1.3 Graphs of Functions . . . . . . . . . . . . . . . . . . . . . 13
Section 1.4 Shifting, Reflecting, and Stretching Graphs . . . . . . . . . 19
Section 1.5 Combinations of Functions . . . . . . . . . . . . . . . . . 23
Section 1.6 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . 29
Section 1.7 Linear Models and Scatter Plots . . . . . . . . . . . . . . . 36
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
P A R T I
3.
2
2
4
4
6
6
8
8
10
10
(2, 3)
x
y
m = −3m = 1
m = 2
m = 0
5. Slope �rise
run�
3
2
C H A P T E R 1Functions and Their Graphs
Section 1.1 Lines in the Plane
2
You should know the following important facts about lines.
■ The graph of is a straight line. It is called a linear equation.
■ The slope of the line through and is
■ (a) If the line rises from left to right. (b) If the line is horizontal.
(c) If , the line falls from left to right. (d) If m is undefined, the line is vertical.
■ Equations of Lines
(a) Slope-Intercept: (b) Point-Slope:
(c) Two-Point: (d) General:
(e) Vertical: (f ) Horizontal:
■ Given two distinct nonvertical lines
(a) is parallel to if and only if
(b) is perpendicular to if and only if m1 � �1�m2.L2L1
m1 � m2 and b1 � b2.L2L1
L1: y � m1x � b1 and L2: y � m2x � b2
y � bx � a
Ax � By � c � 0y � y1 �y2 � y1
x2 � x1
�x � x1�
y � y1 � m�x � x1�y � mx � b
m < 0,
m � 0,m > 0
m �y2 � y1
x2 � x1
.
�x2, y2��x1, y1�y � mx � b
1. (a) Since the slope is positive, the line rises. Matches
(b) m is undefined. The line is vertical. Matches
(c) The line falls.Matches L1.m � �2.
L3.
L2.
m �23.
Vocabulary Check
1. (a) iii (b) i (c) v (d) ii (e) iv 2. slope
3. parallel 4. perpendicular 5. linear extrapolation
Section 1.1 Lines in the Plane 3
7.
−12
−12 12
4
(−4, 0)
(0, −10)
Slope �0 � ��10�
�4 � 0�
10�4
� �52
9.
Slope is undefined.
−2
−10 2
6
(−6, 4)
(−6, −1)
11. Since y does not change. Three points are�0, 1�, �3, 1�, and ��1, 1�.
m � 0, 13. Since m is undefined, x does not change and the lineis vertical. Three points are and �1, 3�.�1, 2�,�1, 1�,
15. Since y decreases 2 for every unitincrease in x. Three points are and �3, �15�.
�2, �13�,�1, �11�,m � �2, 17. Since increases 1 for every increase of 2
in Three points are and �13, 1�.�9, �1�, �11, 0�,x.m �
12, y
19.
(a) Slope:
y-intercept:
(b)
–4 –3 –2 –1 1 2
3
4
5
(0, 3)
x
y
�0, 3�
m � 5
y � 5x � 3
5x � y � 3 � 0 21.
(a) Slope: undefined
No y-intercept
(b)
–1 1 2 3
–2
–1
1
2
x
y
x �25
5x � 2 � 0 23.
(a) Slope:
-intercept:
(b)
x
53( )
−2 −1 1 2
−1
−2
−3
1
0, −
y
�0, �53�y
m � 0
y � �53
3y � 5 � 0
25.
–2 –1 1 2 3 4
–2
–1
1
2
x
(0, −2)
y
� 2y � 3x � 2 ⇒ 3x � y � 2 � 0
y � 2 � 3�x � 0� 27.
–2 –1 1 2 3 4
–3
–4
–5
–1
1
x
(2, −3)
y
x � 2y � 4 � 0
2y � 4 � �x
y � 3 � �12 x � 1
y � ��3� � �12�x � 2�
45. Using the points and you have
S � 2200�2008� � 4,380,300 � $37,300.
When t � 2008,
S � 2200t � 4,380,300.
S � 28,500 � 2200�t � 2004�
m �32,900 � 28,500
2006 � 2004�
44002
� 2200
�2006, 32,900�,�2004, 28,500� 47.
Slope:
y-intercept:
The graph passes through and rises 1 unitfor each horizontal increase of 2.
�0, �2�
�0, �2�
12
y �12
x � 2
�2y � �x � 4
x � 2y � 4
29.
–4 –2 2 4
–6
–4
–2
2
4
6
x(6, −1)
y
x � 6 � 0
x � 6 31.
horizontal line
3
4
2
1
−1
−2
321−1−2−3
− ,( (
x
y
12
32
y �32
� 0
y �32
� 0�x �12� 33.
−1
−2 4
3
1 � y � �3
5x � 2
� 1y � �3
5�x � 5� � 1
y � 1 �5 � 1
�5 � 5�x � 5�
35. Since both points havethe slope is
undefined.
−4
−10 2
4
x � �8
x � �8,37.
−1
−2 4
3
y � �1
2x �
3
2
y � �1
2�x � 2� �
1
2
y �1
2�
54 �
12
12 � 2
�x � 2�
4 Chapter 1 Functions and Their Graphs4 Chapter 1 Functions and Their Graphs
39.
−6
−9 9
6
y � �65
x �1825
y �35
� �65�x �
110�
y �35
��
95 �
35
910 �
110�x �
110�
41.
−2
−3 3
2
� 0.6y � 0.4x � 0.2
� 0.6y � 0.4�x � 1� � 0.6
y � 0.6 ��0.6 � 0.6
�2 � 1�x � 1� 43. The slope is
y � 2x � 5
y � 3 � 2x � 2
y � ��3� � 2�x � 1�
�3 � ��7�1 � ��1� �
42
� 2.
Section 1.1 Lines in the Plane 5
49.
slope is undefined.
no y-intercept
The line is vertical and passes through ��6, 0�.
x � �6 51.
The second setting shows the x- and y-intercepts more clearly.
−4
−2 10
1
−1
−5 10
10
y � 0.5x � 3
53.
and are perpendicular.
−4
−12 12
12
(0, 3)
(0, −1)
(4, 1)
(5, 9)
L2L1
mL2�
1 � 3
4 � 0� �
1
2� �
1
mL1
mL1�
9 � 1
5 � 0� 2 55.
and are parallel.
−8
−12 12
8
(0, −1)
(−6, 0)
(3, 6)
5, 73( (
L2L1
mL2�
73 � 1
5 � 0�
2
3� mL1
mL1�
0 � 6
�6 � 3�
2
3
57.
Slope:
(a)
(b)
1 � y � �12 x � 2
y � 1 � �12 �x � 2�
1 � y � 2x � 3
y � 1 � 2�x � 2�
m � 2
y � 2x �32
4x � 2y � 3 59.
Slope:
(a)
(b)
y �43x �
12772
y �78 �
43�x �
23�
y � �34x �
38
y �78 � �
34�x �
23�
m � �34
y � �34x �
74
3x � 4y � 7 61. vertical line
slope not defined
(a) passes through
(b) passes throughand is horizontal.�3, �2�
y � �2
�3, �2�x � 3 � 0
x � 4 � 0
63. The slope is 2 and lies on the line.Hence,
y � 2x � 1.
y � 1 � 2�x � 1�
y � ��1� � 2�x � ��1��
��1, �1� 65. The slope of the given line is 2. Then has slopeHence,
y � �12x � 1.
y � 2 � �12�x � 2�
y � 2 � �12�x � ��2��
�12.
l
67. (a) (b) (c)
(b) and (c) are perpendicular.
−10
−15 15
10y = 2x
y = − 2xy = x12
y �12xy � �2x y � 2x 69. (a) (b)
(c)
(a) and (b) are parallel.
(c) is perpendicular to (a) and (b).
−10
−15 15
10y = 2x − 4
y = − x12
y = − x + 312
y � 2x � 4
y � �12x � 3y � �
12x
83. (a)
(b)
00
10
25,000
V � �2300t � 25,000
V � 25,000 � �2300t
V � 25,000 �2000 � 25,000
10 � 0�t � 0�
�0, 25,000�, �10, 2000� (c)
etc.
V � �2300�1� � 25,000 � 22,700t � 1:
V � �2300�0� � 25,000 � 25,000t � 0:
t 0 1 2 3 4 5 6 7 8 9 10
V 25,000 22,700 20,400 18,100 15,800 13,500 11,200 8900 6600 4300 2000
6 Chapter 1 Functions and Their Graphs
73.
The maximum height in the attic is 12 feet.
x � 12
4x � 48
34
�x
16
riserun
�34
�x
12�32�
75.
V � 125t � 1790
V � 2540 � 125�t � 6�
�6, 2540�, m � 125
77.
V � �2000t � 32,400
V � 20,400 � �2000�t � 6�
�6, 20,400�, m � �2000 79. The slope is Thisrepresents the decrease in theamount of the loan each week. Matches graph (b).
m � �10. 81. The slope is Thisrepresents the increase in travel cost for each mile driven. Matches graph (a).
m � 0.35.
71. (a) Years Slope
1995–1996
1996–1997
1997–1998
1998–1999
1999–2000
2000–2001
2001–2002
2002–2003
2003–2004
Greatest increase: 1998–1999
Greatest decrease: 1999–2000 ��0.78��0.86�
0.31 � 0.00 � 0.31
0.00 � 0.20 � �0.20
0.20 � 0.92 � �0.72
0.92 � 0.82 � 0.10
0.82 � 1.60 � �0.78
1.60 � 0.74 � 0.86
0.74 � 0.57 � 0.17
0.57 � 0.69 � �0.12
0.69 � 0.91 � �0.22
(b)
(c) Between 1995 and 2004, the earnings per sharedecreased at the rate of 0.07 per year.
(d) For 2010, andwhich is
reasonable.y � �0.07�20� � 1.24 � �0.16,
x � 20
y � �0.07x � 1.24
y � �115
�x � 5� �91100
� �115
x �373300
y � 0.91 �0.31 � 0.91
14 � 5 �x � 5�
�14, 0.31�:�5, 0.91�,
Section 1.1 Lines in the Plane 7
85. (a)
(c)
P � 10.25t � 36,500
P � 27t � �16.75t � 36,500�
P � R � C
� 16.75t � 36,500
C � 36,500 � 5.25t � 11.50t
87. (a) students per year
(b) 1984: students
1997: students
2000: students
(Answers could vary.)
(c) Let represent 1990.
The slope 341 represents the annual increase in students. It is positive, indicatingthat Penn State University increased its students from 1991 to 2005.
y � 341t � 75,008
y �4775
14�t � 1� � 75,349
y � 75,349 �80,124 � 75,349
15 � 1�t � 1�
�1, 75,349�, �15, 80,124�
t � 0
75,349 � 341�9� � 78,418
75,349 � 341�6� � 77,395
75,349 � 341�7� � 72,962
80,124 � 75,3492005 � 1991
�477514
� 341
(b)
(d)
t � 3561 hours
36,500 � 10.25t
0 � 10.25t � 36,500
R � 27t
89. False. The slopes are different:
7 � 4
�7 � 0� �
117
4 � 2
�1 � 8�
27
91.
are the x- and y-intercepts.
−5
−3 9
3
a � 5 and b � �3
�3x � 5y � 15 � 0
x
5�
y
�3� 1
93.
intercepts: �4, 0�, �0, �23�
�2x � 12y � �8
�2
3x � 4y �
�8
3
5
−2
−1
2 x
4�
y
�23
� 1
8 Chapter 1 Functions and Their Graphs
■ Given a set or an equation, you should be able to determine if it represents a function.
■ Given a function, you should be able to do the following.
(a) Find the domain.
(b) Evaluate it at specific values.
1. Yes, it does represent a function. Each domainvalue is matched with only one range value.
3. No, it does not represent a function. The domainvalues are each matched with three range values.
7. No, it does not represent a function. The input values of 10 and 7 are each matched with two output values.
9. (a) Each element of A is matched with exactly one element of B, so it does represent a function.
(b) The element 1 in A is matched with two elements, and 1 of B, so it does not represent a function.
(c) Each element of A is matched with exactly one element of B, so it does represent a function.
(d) The element 2 of A is not matched to any element of B, so it does not represent a function.
�2
Section 1.2 Functions
5. Yes, the relation represents y as a function of x.Each domain value is matched with only onerange value.
Vocabulary Check
1. domain, range, function 2. independent, dependent 3. piecewise-defined
4. implied domain 5. difference quotient
95.
3x � 2y � 6 � 0
x � 6x
2�
y
3� 1 97.
12x � 3y � 2 � 0
�6x �32
y � 1
x
�1�6�
y�2�3
� 1 99. The slope is positive andthe y-intercept is positive.Matches (a).
101. Both lines have positiveslope, but their y-interceptsdiffer in sign. Matches (c).
103. No. The line does nothave an x-intercept.
y � 2 105. Yes. Answers will vary.
107. Yes. x � 20 109. No. The term
causes the expression to not be a polynomial.
x�1 �1x
111. No. This expression is notdefined for x � ±3.
117. Answers will vary.
113. x2 � 6x � 27 � �x � 9��x � 3� 115. 2x2 � 11x � 40 � �2x � 5��x � 8�
Section 1.2 Functions 9
11. Each are functions. For each year there correspondsone and only one circulation.
13.
Thus, y is not a function of x. For instance, thevalues both correspond to x � 0.y � 2 and �2
x2 � y2 � 4 ⇒ y � ±�4 � x2
15.
This is a function of x.
y � �x2 � 1 17.
Thus, y is a function of x.
2x � 3y � 4 ⇒ y �13�4 � 2x�
19.
Thus, y is not a function of x. For instance, thevalues and both correspond to x � 2.��3y � �3
y2 � x2 � 1 ⇒ y � ±�x2 � 1 21.
This is a function of x.
y � �4 � x�
23. does not represent as a function of All values of correspond to x � �7.yx.yx � �7
25.
(a)
(c) f�4t� �1
�4t� � 1�
1
4t � 1
f�4� �1
�4� � 1�
1
5
f�x� �1
x � 1
(b)
(d) f�x � c� �1
�x � c� � 1�
1
x � c � 1
f �0� �1
�0� � 1� 1
27.
(a)
(b)
(c) f�t � 2� � 3�t � 2� � 1 � 3t � 7
f��4� � 3��4� � 1 � �11
f�2� � 3�2� � 1 � 7
f�t� � 3t � 1
31.
(a)
(b)
(c) f�4x2� � 3 � �4x2 � 3 � 2�x�f �0.25� � 3 � �0.25 � 2.5
f�4� � 3 � �4 � 1
f�y� � 3 � �y
29.
(a)
(b)
(c) h�x � 2� � �x � 2�2 � 2�x � 2� � x2 � 2x
h�1.5� � �1.5�2 � 2�1.5� � �0.75
h�2� � 22 � 2�2� � 0
h�t� � t2 � 2t
35.
(a)
(b)
(c)
is undefined.f �0�
f�t� � �t�t
� �1�1
if t > 0if t < 0
f��3� � ��3��3
� �1
f�3� � �3�3
� 1
f�x� � �x�x
37.
(a)
(b)
(c) f�2� � 2�2� � 2 � 6
f�0� � 2�0� � 2 � 2
f��1� � 2��1� � 1 � �1
f�x� � �2x � 1, x < 0
2x � 2, x ≥ 0
33.
(a)
(b)
(c) q�y � 3� �1
�y � 3�2 � 9�
1
y2 � 6y
q�3� �1
32 � 9 is undefined.
q�0� �1
02 � 9� �
1
9
q�x� �1
x2 � 9
63.
Domain:
Range: �0, 2�
��2, 2�
−4
−6 6
4
f �x� � �4 � x2 65.
Domain:
Range: �0, ��
���, ��
−2
−8 4
6
g�x� � �2x � 3�
67.
��2, 4�, ��1, 1�, �0, 0�, �1, 1�, �2, 4�
f�x� � x2 69.
��2, 4�, ��1, 3�, �0, 2�, �1, 3�, �2, 4�
f �x� � �x� � 2
10 Chapter 1 Functions and Their Graphs
49.
x �43
3x � 4
3x � 4 � 0
f �x� �3x � 4
5� 0 51.
x � �1 or x � 2
�x � 1��x � 2� � 0
x2 � x � 2 � 0
x2 � x � 2
f �x� � g�x�
53.
Since is a polynomial, the domain is all realnumbers x.
f�x�
f�x� � 5x2 � 2x � 1 55.
Domain: All real numbers except t � 0
h�t� �4
t
57.
Domain: all real numbers
f �x� � 3�x � 4 59.
Domain: All real numbers except
x � 0, x � �2
g�x� �1
x�
3
x � 2
61.
Domain: all y > 10.
y > 10
y � 10 > 0
g�y� �y � 2
�y � 10
47.
x � 5
3x � 15
f �x� � 15 � 3x � 0
43. h�t� �12�t � 3�
41.
(a)
(b)
(c) f�4� � 42 � 1 � 17
f�1� � 4
f��2� � ��2� � 2 � 0
f�x� � �x � 2,4,x2 � 1,
x < 0 0 ≤ x < 2 x ≥ 2
t
1 0 112
12h�t�
�1�2�3�4�5
45. f�x� � ��12x � 4, x ≤ 0
�x � 2�2, x > 0
x 0 1 2
5 4 1 092f �x�
�1�2
39.
(a)
(b)
(c) f �2� � 2�2�2 � 2 � 10
f �1� � �1�2 � 2 � 3
f ��2� � ��2�2 � 2 � 6
f �x� � �x2 � 2,
2x2 � 2,
x ≤ 1
x > 1
Section 1.2 Functions 11
71.
�C2
4�A � �� C
2��2
r �C
2�
A � �r2 , C � 2�r
73. (a) According to the table, the maximum profit is 3375 for
(b) Yes, P is a function of x.
(c)
P � �30x,45x � 0.15x2,
x ≤ 100x > 100
� 45x � 0.15x2, x > 100
� �105 � 0.15x�x � 60x
� �90 � �x � 100��0.15��x � 60x
� �price per unit��number of units� � �cost��number of units�
Profit � Revenue � Cost
3100100 180
3400
x � 150.
75.
Since and all lie on the same line, the slopes between any pair of points are equal.
Therefore,
The domain is since A > 0.x > 2,
A �12
xy �12
x� xx � 2� �
x2
2x � 4.
y � 1 �2
2 � x�
xx � 2
1 � y �2
2 � x
1 � y2 � 0
�1 � 02 � x
�x, 0��0, y�, �2, 1�
A �12�base��height� �
12xy.
77. (a)
But, or
Thus,
Domain: 0 < x < 27
x < 27.
4x < 108
Since y � 108 � 4x > 0
V � �108 � 4x�x2.
y � 108 � 4x.y � 4x � 108,
V � �length��width��height� � yx2 (b)
(c) The highest point on the graph occurs atThe dimensions that maximize the
volume are 18 � 18 � 36 inches.x � 18.
00 27
12,000
12 Chapter 1 Functions and Their Graphs
85. (a)
(Answers will vary.)
F increases very rapidly as y increases.
(b)
00 50
5,000,000
F�y� � 149.76�10y5�2
y 5 10 20 30 40
4.79 � 1062.33 � 1068.47 � 1051.50 � 1052.65 � 104F�y�
(c) From the table, (slightly above 20). Youcould obtain a better approximation by completingthe table for values of y between 20 and 30.
(d) By graphing together with the horizontal lineyou obtain feet.y 21.37y2 � 1,000,000,
F�y�
y 22 ft
87.
�2cc
� 2, c � 0
f �x � c� � f �x�
c�
2�x � c� � 2xc
f �x� � 2x 89.
�h2 � 3h
h� h � 3, h � 0
�4 � 4h � h2 � 2 � h � 1 � 3
h
f �2 � h� � f �2�
h�
�2 � h�2 � �2 � h� � 1 � 3h
f �x� � x2 � x � 1, f �2� � 3
12 Chapter 1 Functions and Their Graphs
79. The domain of is
The domain of is
You can tell by comparing the models to the givendata. The models fit the data well on the domainsabove.
1 ≤ x ≤ 6.0.505x2 � 1.47x � 6.3
7 ≤ x ≤ 12.�1.97x � 26.3 81.
$4,630 in monthly revenue for November.
f �11� � �1.97�11� � 26.3 � 4.63
83.
corresponds to 1990.t � 0
n�t� � ��6.13t2 � 75.8t � 577,24.9t � 672,
0 ≤ t ≤ 66 < t ≤ 13
t 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Model 577 647 704 749 782 803 811 846 871 896 921 946 971 996
91.
�1 � t
t�t � 1� ��1
t, t � 1
f �t� � f �1�t � 1
�
1t
� 1
t � 1
f �t� �1t, f �1� � 1 93. False. The range of f �x� is ��1, ��.
95. f�x� � �x � 4,4 � x2,
x ≤ 0x > 0
97. f�x� � �2 � x,4,x � 1,
x ≤ �2�2 < x < 3x ≥ 3
Section 1.3 Graphs of Functions 13
Section 1.3 Graphs of Functions
■ You should be able to determine the domain and range of a function from its graph.
■ You should be able to use the vertical line test for functions.
■ You should be able to determine when a function is constant, increasing, or decreasing.
■ You should be able to find relative maximum and minimum values of a function.
■ You should know that f is
(a) Odd if
(b) Even if f��x� � f�x�.f��x� � �f�x�.
1. Domain: All real numbers
Range:
f �0� � 1
���, 1]
3. Domain:
Range:
f �0� � 4
�0, 4�
��4, 4�
Vocabulary Check
1. ordered pairs 2. Vertical Line Test 3. decreasing
4. minimum 5. greatest integer 6. even
99. The domain is the set of inputs of the function and the range is the set of corresponding outputs.
101. 12 �4
x � 2�
12�x � 2� � 4x � 2
�12x � 20
x � 2
103.
��x � 6��x � 10�
5�x � 3� , x � 0, 12
��2x � 1��x � 6��x � 10�
5�2x � 1��x � 3�
2x3 � 11x2 � 6x5x
�x � 10
2x2 � 5x � 3�
x�2x2 � 11x � 6��x � 10�5x�2x � 1��x � 3�
9.
Domain: All real numbers
Range: �0, ��
−1
−9 3
7
f�x� � �x � 3�5.
Domain: All real numbers
Range: �3, ��
−1
−6 6
7
f �x� � 2x2 � 3 7.
Domain:or
Range: �0, ��
�1, ��x � 1 ≥ 0 ⇒ x ≥ 1
−1
−1 5
3
f �x� � �x � 1
27.
(a)
(b) Increasing on
Decreasing on ��3, �2�
��2, ��
−3
−9 9
9
f�x� � x�x � 3
14 Chapter 1 Functions and Their Graphs
15.
A vertical line intersects the graph just once, so y isa function of x. Graph y1 �
12x2.
y �12x2 17.
A vertical line intersects the graph more than once,so y is not a function of x. Graph the circle as
y2 � ��25 � x2.
y1 � �25 � x2
x2 � y2 � 25
19.
f is increasing on ���, ��.
f�x� �32 x 21.
f is increasing on
f is decreasing on �0, 2�.
���, 0� and �2, ��.
f�x� � x3 � 3x2 � 2
23.
(a)
(b) f is constant on ���, ��.
−2
−6 6
6
f �x� � 3 25.
(a)
(b) Increasing on
Decreasing on ���, 0�
�0, ��
−2
−6 6
6
f �x� � x2�3
11.
(a) Domain: all real numbers
(b)
(c) These are the x-intercepts of f.
(d)
(e) This is the y-intercept of f.
(f) The coordinates are
(g) The coordinates are
(h) ��3, f ��3�� � ��3, 6�.f ��3� � ��3�2 � ��3� � 6 � 6.
��1, �4�.f ��1� � ��1�2 � ��1� � 6 � �4.
�1, �6�f �1� � 12 � 1 � 6 � �6.
f �0� � �6
f �x� � x2 � x � 6 � �x � 3��x � 2� � 0 ⇒ x � 3, �2
f �x� � x2 � x � 6
13.
(a) Domain: all
(b)
(c) x-intercepts
(d)
(e) y-intercept
(f)
(g)
(h) f ��3� � ��3 � 1� � 2 � 2, ��3, 2�
f ��1� � ��1 � 1� � 2 � 0, ��1, 0�
f �1� � �1 � 1� � 2 � �2, �1, �2�
f �0� � �0 � 1� � 2 � �1
�x � 1� � 2 � 0 ⇒ �x � 1� � 2 ⇒ x � �1, 3
x
f �x� � �x � 1� � 2
Section 1.3 Graphs of Functions 15
29.
(a)
(b) Increasing on constant on decreasing on ���, �1�
��1, 1�,�1, ��,
−2
−6 6
6
f �x� � �x � 1� � �x � 1� 31.
Relative minimum:
−10
−6 12
2
�3, �9�
f�x� � x2 � 6x 33.
Relative minimum:
Relative maximum:
−8
−6 6
24
��2, 20�
�1, �7�
y � 2x3 � 3x2 � 12x
35.
Relative minimum:
is not a relative maximum because it occurs at the endpoint of the domain �0, ��.�0, 0�
−1
−1 5
3
�0.33, �0.38�
h�x� � �x � 1��x
37.
(a)
Minimum: �2, �9�
−2−4 2 4 6 8 10 12 14 16
−4
−6
−8
−10
6
8
x
y
f(x) = x2 − 4x − 5
(2, −9)
f �x� � x2 � 4x � 5
(b)
Minimum:
(c) Answers are the same.
�2, �9�
−18
−12
18
12
39.
(a)
Relative maximum:
Relative minimum:
(b) Relative maximum:
Relative minimum:
(c) Answers are the same.
�1, �2�
��1, 2�
�1, �2�
��1, 2�
f(x) = x3 − 3x(−1, 2)
(1, −2)
−1−3 1 2 3 4 5 6 7
−2
−3
−4
−5
1
3
4
5
x
y
f �x� � x3 � 3x 41.
(a)
Relative minimum:
(b) Relative minimum:
(c) Answers are the same.
�1, �2�
�1, �2�
f(x) = 3x2 − 6x + 1
(1, −2)
−1−3−4−5−6−7 −2 1 2 3
−2
−3
1
2
3
x
y
f �x� � 3x2 � 6x � 1
71.
(a) If f is even, another point is
(b) If f is odd, another point is ��x, y�.
��x, �y�.
�x, �y�
16 Chapter 1 Functions and Their Graphs
55.
–5 –4 –3 –2 –1 1 2 3 4
–5
–4
1
2
3
4
x
y
f �x� � �2x 57.
Domain:
Range:
Sawtooth pattern
�0, 2�
���, ��
−4
−9 9
8
s�x� � 2�14x � �1
4x� 59.
f is neither even nor odd.
� f�t� � �f�t�
� t2 � 2t � 3
f��t� � ��t�2 � 2��t� � 3
63.
The function is odd.
� �f �x�
� �x�1 � x2
f ��x� � ��x��1 � ��x�2 65.
The function is even.
� g �s�
� 4s2�3
g ��s� � 4 ��s�2�361.
g is odd.
� �g�x�
� �x3 � 5x
g ��x� � ��x�3 � 5��x�
67.
(a) If f is even, another point is
(b) If f is odd, another point is �32, �4�.�3
2, 4�.��3
2, 4� 69.
(a) If f is even, another point is
(b) If f is odd, another point is ��4, �9�.
��4, 9�.
�4, 9�
16 Chapter 1 Functions and Their Graphs
43.
−1−3 1 2 3 4 5−1
−2
−3
−4
1
3
4
x
y
f�x� � 2x � 3, x < 0
23 � x, x ≥ 045.
x
y
−1−2−3−4 1 2 3 4−1
−2
−3
1
3
4
5
f �x� � �x � 4,�4 � x,
x < 0x ≥ 0
47.
x
y
−1−2−4 1 2 3 4−1
−2
−3
1
3
4
5
f �x� � x � 3,3,2x � 1,
x ≤ 00 < x ≤ 2x > 2
49.
x
y
−1−2−3−4 2 3 4
−3
−4
−5
1
2
3
f �x� � 2x � 1,x2 � 2,
x ≤ �1x > �1 51.
–5 –4 –1 1 2 3 4
–3
–2
2
3
4
5
6
x
y
f �x� � �x � 2 53.
–4 –3 1 2 3 4 5
–3
–2
2
1
3
4
5
6
x
y
f �x� � �x � 1 � 2
Section 1.3 Graphs of Functions 17
73.
−4
−9 9
8
f �x� � 5, even 75. is neither even nor odd.
−6
−9 9
6
f �x� � 3x � 2 77.
−6
−6 6
2
h �x� � x2 � 4, even
79. is neither evennor odd.
−1
−4 2
3
f �x� � �1 � x 81. is neither evennor odd.
−1
−5 1
3
f �x� � �x � 2� 83.
–1 1 2 3 4 5
1
−1
2
3
4
5
x
y
���, 4�
4 ≥ x
f�x� � 4 � x ≥ 0
85.
–6 –4 2 4 6
–10
–4
–2
2
x
y
�3, �� or ���, �3�
x ≥ 3 or x ≤ �3
x2 ≥ 9
f�x� � x2 � 9 ≥ 0 87. (a) The second model is correct. For instance,
� 1.05 � 0.38��12�� � 1.05.
C2�12� � 1.05 � 0.38� ���1
2 � 1���
(b)
The cost of an 18-minute 45-second call is
� 1.05 � 0.38�18� � $7.89.
� 1.05 � 0.38���17.75�� � 1.05 � 0.38��18�
C2�184560� � C2�18.75� � 1.05 � 0.38����18.75 � 1���
00 60
25
89.
� �x2 � 4x � 3, 1 ≤ x ≤ 3
� ��x2 � 4x � 1� � 2
h � top � bottom
18 Chapter 1 Functions and Their Graphs
93. False. The domain of is the set of all real numbers.f �x� � �x2
95. c 97. b 99. a
101.
Therefore, is odd.f �x�
� �a2n�1x2n�1 � a2n�1x
2n�1 � . . . � a3x3 � a1x � �f �x�
f��x� � a2n�1��x�2n�1 � a2n�1��x�2n�1 � . . . � a3��x�3 � a1��x�
f�x� � a2n�1x2n�1 � a2n�1x
2n�1 � . . . � a3x3 � a1x
103. f is an even function.
(a) is even because
(c) is even becauseg��x� � f ��x� � 2 � f �x� � 2 � g�x�.g�x� � f �x� � 2
g��x� � �f ��x� � �f �x� � g�x�.g�x� � �f �x� (b) is even because
(d) is neither even nor odd becausenor
�g�x�.g��x� � �f ��x � 2� � �f �x � 2� � g�x�g�x� � �f �x � 2�
g��x� � f ����x�� � f �x� � f ��x� � g�x�.g�x� � f ��x�
105. No, does notrepresent x as a function of y.For instance, and
both lie on the graph.�3, 4���3, 4�
x2 � y2 � 25 107.
Terms:
Coefficients: �2, 8
�2x2, 8x
�2x2 � 8x 109.
Terms:
Coefficients:13
, �5, 1
x3
, �5x2, x3
x3
� 5x2 � x3
18 Chapter 1 Functions and Their Graphs
91.
(a) 1850
1417500
0 ≤ t ≤ 14
P�t� � 0.0108t4 � 0.211t3 � 0.40t2 � 7.9t � 1791
(b) P is increasing from 1990 to 1995
and from 2001 to 2004. P is decreasingfrom 1995 to 2001.
(c) The maximum population was about 1,821,000 in1995 �t � 5.7�.
�t � 11.8��t � 5.7�,�t � 0�
111. (a)
(b) midpoint � ��2 � 62
, 7 � 3
2 � �2, 5�
� �64 � 16 � �80 � 4�5
d � ��6 � ��2��2 � �3 � 7�2 113. (a)
(b) midpoint � �52
�32
2,
�1 � 42
� �12
, 32
� �16 � 25 � �41
d ����32
�52
2
� �4 � ��1��2
115.
(a)
(b)
(c) f�x � 3� � 5�x � 3� � 1 � 5x � 16
f��1� � 5��1� � 1 � �6
f�6� � 5�6� � 1 � 29
f �x� � 5x � 1 117.
(a)
(b)
(c) f�6� � 6�6 � 3 � 6�3
� 12�3� � 36 � 12�9
f�12� � 12�12 � 3
f�3� � 3�3 � 3 � 0
f �x� � x�x � 3
Section 1.4 Shifting, Reflecting, and Stretching Graphs 19
Section 1.4 Shifting, Reflecting, and Stretching Graphs
■ You should know the graphs of the most commonly used functions in algebra, and be able to reproducethem on your graphing utility.
(a) Constant function: (b) Identity function:
(c) Absolute value function: (d) Square root function:
(e) Squaring function: (f ) Cubing function:
■ You should know how the graph of a function is changed by vertical and horizontal shifts.
■ You should know how the graph of a function is changed by reflection.
■ You should know how the graph of a function is changed by nonrigid transformations, like stretches andshrinks.
■ You should know how the graph of a function is changed by a sequence of transformations.
f�x� � x3f�x� � x2
f�x� � �xf�x� � �x�f�x� � xf�x� � c
1.
−4−6 −2
2
2
4
4
6
6
f (x)h (x)
g(x)
x
y 3.
−4 −2−6 2 4 6
f (x)
h(x)
g(x)
−2
4
x
y 5.
−4
−4 −2
−6
−6
4
2
4 6
f(x) h(x)g(x)
x
y
Vocabulary Check
1. quadratic function 2. absolute value function 3. rigid transformations
4. 5. 6. (a) ii (b) iv (c) iii (d) ic > 1, 0 < c < 1�f �x�, f ��x�
119.
h � 0f�3 � h� � f�3�
h�
�h2 � 4h � 12� � 12h
�h�h � 4�
h� h � 4,
f�3� � 32 � 2�3� � 9 � 12
� h2 � 4h � 12
f�3 � h� � �3 � h�2 � 2�3 � h� � 9 � 9 � 6h � h2 � 6 � 2h � 9
f �x� � x2 � 2x � 9
11.
−2
−2
2
42
4
6
6
f x( )
h x( )
g x( )
x
y7.8
6
4
−2
−4
642−2−4−6
f x( )h x( )
g x( )
x
y 9.
−4
−4−6 −2 2
2
4
6
8
6
f(x)
h(x)g(x)
−2
x
y
33. is obtained from by a reflectionin the x-axis followed by a vertical shift upward offour units.
f�x�g�x� � 4 � x3
35. is obtained from by a leftshift of two units and a vertical shrink by a factorof 1
4.
f�x�h�x� �14�x � 2�3 37. is obtained from by a
horizontal stretch followed by a vertical shift two units upward.
f �x�p�x� � �13 x�3
� 2
20 Chapter 1 Functions and Their Graphs
15. Horizontal shift three units to left of (or vertical shift three units upward)y � x � 3
y � x: 17. Vertical shift one unit downward of
y � x2 � 1
y � x2
19. Reflection in the x-axis and a vertical shift one unitupward of y � �x: y � 1 � �x
21. is reflected in the x-axis,followed by a vertical shift one unit downward.
f �x�y � ��x � 1
23. shifted right two units.y � �x � 2 is f �x� 25. is a vertical stretch of f �x� � �x.y � 2�x
27. is shifted left five units.f �x�y � �x � 5� 29. is reflected in the x-axis.f�x�y � ��x�
31. is a vertical stretch of f �x�.y � 4�x�
13. (a)
y
x21 3 4 5
4
3
2
1
5
(0, 1)
(1, 2)
(3, 3)
(4, 4)
y
y � f �x� � 2 (b)
y
x1 3 4 5
2
1
−1
−2
−3
(0, 1)
(1, 0)
(3, −1)
(4, −2)
y
y � �f �x� (c)
x2 3 5 61 4
1
2
3
4
−1
−2(2, −1)
(3, 0)
(5, 1)
(6, 2)
y
y � f �x � 2�
(g) Let Then from the graph,
g�8� � f �12�8�� � f �4� � 2
g�6� � f �12�6�� � f �3� � 1
g�2� � f �12�2�� � f �1� � 0
g�0� � f �12�0�� � f �0� � �1
g�x� � f �12x�.
x
y
2 3 4 5 6 7 8−1
−2
−3
−4
1
2
3
4
5
(0, −1)
(2, 0) (6, 1)
(8, 2)
(d)
y
x1 2−13−
2
3
−1
−2
(0, 1)
(1, 2)
(−2, 0)
(−3, −1)
y
y � f �x � 3� (e)
5
4
3
2
1
−2
−3
54321−2 −1−3
(4, 4)
(3, 2)
(0, 2)−
(1, 0)
x
y
y � 2 f �x� (f)
y
x−1−3 −2−4−5
2
3
−2
(0, −1)
(−1, 0)(−3, 1)
(−4, 2)
y
y � f ��x�
Section 1.4 Shifting, Reflecting, and Stretching Graphs 21
39.
is ahorizontal shift two units to left.
is a vertical shrink.h �x� �12 f �x� �
12�x3 � 3x2�
g�x� � f �x � 2� � �x � 2�3 � 3�x � 2�2
f �x� � x3 � 3x2
41.
reflection in the x-axis and vertical shrink
reflection in the y-axish�x� � f ��x� � ��x�3 � 3��x�2
g�x� � �13 f�x� � �
13�x3 � 3x2�
−4
−6 6
4
fh g
f�x� � x3 � 3x2
43. (a)
(b) is obtained from by ahorizontal shift to the left five units, a reflectionin the -axis, and a vertical shift upward twounits.
(c)
(d) g�x� � 2 � f �x � 5�
3
2
1
−2
−3
−4
−5
−6
−7
1−2 −1−4−5−9 −7−8x
y
x
fg�x� � 2 � �x � 5�2
f �x� � x2
−4
−5 7
4
f
hg
45. (a)
(b) is obtained from by ahorizontal shift four units to the right, a verticalstretch of 2, and a vertical shift upward threeunits.
(c)
(d) g�x� � 3 � 2 f �x � 4�
7
6
5
4
3
2
1
−1 7654321x
y
fg�x� � 3 � 2�x � 4�2
f �x� � x2
47. (a)
(b) is obtained from by ahorizontal shift two units to the right followed by a vertical stretch of 3.
(c)
(d) g�x� � 3f �x � 2�
3
2
1
−1
−2
−3
54321−1x
y
fg�x� � 3�x � 2�3
f �x� � x3 49. (a)
(b) is obtained from by ahorizontal shift one unit to the right, and avertical shift upward two units.
(c)
(d) g�x� � f �x � 1� � 2
5
4
3
2
1
54321−1−2−3x
y
fg�x� � �x � 1�3 � 2
f �x� � x3
63. (a) is a reflection in the y-axis, so the graph is increasing on and decreasing on
(b) is a reflection in the x-axis, so the graph is decreasing on and increasingon
(c) is a vertical stretch, so the graph is increasing on and decreasingon
(d) is a vertical shift, so the graph is increasing on and decreasing on
(e) is a horizontal shift one unit tothe left, so the graph is increasing on and decreasing on �1, ��.
���, 1�y � f �x � 1�
�2, ��.���, 2�
y � f �x� � 3
�2, ��.���, 2�
y � 2 f �x�
�2, ��.���, 2�
y � �f �x�
��2, ��.���, �2�
y � f ��x�
22 Chapter 1 Functions and Their Graphs
57. (a) is a vertical stretch offollowed by a vertical shift of 33.0.
(b)
(c)
corresponds to 1990.
corresponds to 2003.G�0� � F�13�
G��13� � F�0�
�13 ≤ t ≤ 0.
G�t� � F�t � 13� � 33.0 � 6.2�t � 13,
60
130
0
f �t� � �t,F�t� � 33.0 � 6.2�t
59. False. is a reflection in the y-axis.y � f ��x�
61. (a) is a reflection in the y-axis, so the x-intercepts are and
(b) is a reflection in the x-axis, so the x-intercepts are and
(c) is a vertical stretch, so the x-intercepts are the same: ,
(d) is a vertical shift, so you cannotdetermine the x-intercepts.
(e) is a horizontal shift 3 units to theright, so the x-intercepts are and x � 0.x � 5y � f �x � 3�
y � f �x� � 2
�3.x � 2y � 2 f �x�
x � �3.x � 2y � �f �x�
x � 3.x � �2y � f ��x�
22 Chapter 1 Functions and Their Graphs
51. (a)
(b) is obtained from f by a horizontal shift four units to the left, followedby a vertical shift eight units upward.
(c)
(d) g�x� � f �x � 4� � 8
x
y
−4−8−12−16 4 8 12−4
4
8
12
16
20
24
g�x� � �x � 4� � 8
f �x� � �x� 53. (a)
(b) is obtained from f by ahorizontal shift one unit to the right, a verticalstretch of 2, a reflection in the x-axis, and avertical shift downward four units.
(c)
(d) g�x� � �2f �x � 1� � 4
x
y
−2−4−6−8 2 4 6 8−2
−4
−6
−12
−14
2
g�x� � �2�x � 1� � 4
f �x� � �x�
55. (a)
(b) is obtained from f by ahorizontal shift three units to the left, a verticalshrink, a reflection in the x-axis, and a verticalshift one unit downward.
(c)
(d) g�x� � �12 f �x � 3� � 1
5
4
3
1
2
−3
−4
−5
54321−2−4−5x
y
g�x� � �12�x � 3 � 1
f �x� � �x
Section 1.5 Combinations of Functions 23
Section 1.5 Combinations of Functions
■ Given two functions, f and g, you should be able to form the following functions (if defined):
1. Sum:
2. Difference:
3. Product:
4. Quotient:
5. Composition of f with
6. Composition of g with f : �g � f ��x� � g� f�x��g: � f � g��x� � f�g�x��
� f�g��x� � f�x��g�x�, g�x� � 0
� fg��x� � f�x�g�x�� f � g��x� � f�x� � g�x�
� f � g��x� � f�x� � g�x�
1.
x
y
−1−2 1 2 3 4−1
−2
1
2
3
4
h
3.
x
y
−1−2−3 1 2 3 4 5
1
2
4
5
6
7
h
Vocabulary Check
1. addition, subtraction, multiplication, division 2. composition
3. 4. inner, outerg�x�
65. The vertex is approximately at and the graphopens upward. Matches (c).
�2, 1� 67. The vertex is approximately and the graphopens upward. Matches (c).
�2, �4�
69. Slope
Slope
Neither parallel nor perpendicular
L2: 9 � 33 � 1
�32
L1: 10 � 22 � 2
� 3 71. Domain: All x � 9
73. Domain: ⇒ �10 ≤ x ≤ 10100 � x2 ≥ 0 ⇒ x2 ≤ 100
24 Chapter 1 Functions and Their Graphs
17.
� 30
� 15�2�
� �42 � 1��4 � 2�
� fg��4� � f �4�g�4� 19.
� �247
�24�7
���5�2 � 1
�5 � 2
� fg���5� �
f ��5�g ��5�
7.
(a)
(b)
(c)
(d)
Domain: all x � 1.
� f
g� �x� �f �x�g�x�
�x2
1 � x, x � 1
� fg��x� � f�x� � g�x� � x2�1 � x� � x2 � x3
� f � g��x� � f�x� � g�x� � x2 � �1 � x� � x2 � x � 1
� f � g��x� � f�x� � g�x� � x2 � �1 � x� � x2 � x � 1
f �x� � x2, g�x� � 1 � x
9.
(a)
(b)
(c)
(d)
Domain: x < 1
� fg��x� �
x2 � 5�1 � x
� fg��x� � �x2 � 5��1 � x
� f � g��x� � x2 � 5 � �1 � x
� f � g��x� � x2 � 5 � �1 � x
f�x� � x2 � 5, g�x� � �1 � x 11.
(a)
(b)
(c)
(d)
Domain: x � 0
� fg��x� �
1�x1�x2 � x, x � 0
� fg��x� �1x
�1x2 �
1x3
� f � g��x� �1x
�1x2 �
x � 1x2
� f � g��x� �1x
�1x2 �
x � 1x2
f�x� �1x, g�x� �
1x2
5.
(a)
(b)
(c)
(d)
Domain: all x � 3
x � 3� fg��x� �
f �x�g�x� �
x � 3x � 3
,
� fg��x� � f �x�g�x� � �x � 3��x � 3� � x2 � 9
� f � g��x� � f �x� � g�x� � �x � 3� � �x � 3� � 6
� f � g��x� � f �x� � g�x� � �x � 3� � �x � 3� � 2x
g�x� � x � 3 f �x� � x � 3,
13.
� 8 � 1 � 9
� �32 � 1� � �3 � 2�
� f � g�� 3� � f �3� � g�3� 15.
� 1
� �0 � 1� � �0 � 2�
� f � g��0� � f �0� � g�0�
Section 1.5 Combinations of Functions 25
21.
� 4t2 � 2t � 1
� ��2t�2 � 1� � �2t � 2�
� f � g��2t� � f �2t� � g�2t� 23.
� �125t3 � 50t2 � 5t � 2
� �25t2 � 1���5t � 2�
� ���5t�2 � 1���5t � 2�
� fg���5t� � f ��5t�g��5t�
25.
�1 � t2
t � 2, t � �2
�t2 � 1
�t � 2
���t�2 � 1
�t � 2
� fg���t� �
f ��t�g ��t�
29. 4
4
−2
−5h
f
g
33.
contributes more to the magnitude inboth intervals.f �x� � 3x � 2
−6
−9 9
6
f g
f + g
� f � g��x� � 3x � 2 � �x � 5
f �x� � 3x � 2, g �x� � ��x � 5,31.
For , contributes more to themagnitude.
For , contributes more to the magnitude.g�x�x > 6
f�x�0 ≤ x ≤ 2
−10
−14 16
10
fg
f + g
f�x� � 3x, g�x� � �x3
10, � f � g��x� � 3x �
x3
10
27. 3
5
−3
−4
h
f
g
37.
(a)
(b)
(c) � f � g��0� � 20
�g � f ��x� � g � f�x�� � g �3x � 5� � 5 � �3x � 5� � �3x
� f � g��x� � f�g�x�� � f�5 � x� � 3�5 � x� � 5 � 20 � 3x
f�x� � 3x � 5, g�x� � 5 � x
35.
(a)
(b)
(c) � f � g��0� � �0 � 1�2 � 1
�g � f ��x� � g � f�x�� � g �x2� � x2 � 1
� f � g��x� � f�g�x�� � f�x � 1� � �x � 1�2
f�x� � x2, g�x� � x � 1
39. (a) The domain of is or
(b) The domain of is all real numbers.
(c)
The domain of is all real numbers.� f � g�
� f � g��x� � f �g�x�� � f �x2� � �x2 � 4.
g�x� � x2
x ≥ �4.x � 4 ≥ 0f �x� � �x � 4 41. (a) The domain of is all real
numbers.
(b) The domain of is all
(c)
The domain of is x ≥ 0.f � g
� ��x �2 � 1 � x � 1, x ≥ 0
� f � g��x� � f �g�x�� � f ��x �x ≥ 0.g�x� � �x
f �x� � x2 � 1
26 Chapter 1 Functions and Their Graphs
(c)55. (a)
(b) No, because 24 � 5x � �5x.� f � g��x� � �g � f ��x�
�g � f ��x� � g � f �x�� � g �5x � 4� � 4 � �5x � 4� � �5x
� f � g��x� � f �g�x�� � f �4 � x� � 5�4 � x� � 4 � 24 � 5xx
0 24 0
1 19
2 14
3 9 �15
�10
�5
g� f �x��f �g�x��
53. (a)
Domain: all x
(b) They are equal.
−1
−3 3
3
f ° g = g ° f
�g � f ��x� � g� f�x�� � g�x2�3� � �x2�3�6 � x4
� f � g��x� � f �g�x�� � f�x6� � �x6�2�3 � x4
26 Chapter 1 Functions and Their Graphs
47. (a) The domain of is all real numbers.
(b) The domain of is all
(c)
Domain: x � ±2
� f � g��x� � f �g�x�� � f � 1
x2 � 4� �1
x2 � 4� 2
x � ±2g�x� �1
x2 � 4
f �x� � x � 2 49. (a)
Domain: all x
(b) They are not equal.
0−6 6
8
g ° ff ° g
� x � 4, x ≥ �4
�g � f��x� � g� f�x�� � g��x � 4 � � ��x � 4 �2
� f � g��x� � f �g�x�� � f�x2� � �x2 � 4
51. (a)
Domain: all x
The domain of is all real numbers.
(b) They are equal.
f ° g = g ° f−13 7
−6
6
f � g
� 3�13x � 3� � 9 � x
�g � f ��x� � g� f �x�� � g�13x � 3�
� 13�3x � 9� � 3 � x
� f � g��x� � f�g�x�� � f�3x � 9�
43. (a) The domain of is all
(b) The domain of is all real numbers.
(c) The domain of is all x � �3.� f � g��x� � f �x � 3� �1
x � 3
g�x� � x � 3
x � 0.f �x� �1x
45. (a) The domain of is all real numbers.
(b) The domain of is all real numbers.
(c)
Domain: all real numbers
� ��3 � x� � 4� � ��x � 1� � �x � 1�� f � g��x� � f �g�x�� � f �3 � x�
g�x� � 3 � x
f �x� � �x � 4�
Section 1.5 Combinations of Functions 27
(c)57. (a)
(b) No, because �x2 � 1 � x � 1.� f � g��x� � �g � f ��x�
x ≥ �6� x � 1,� �x � 6� � 5
� ��x � 6 �2� 5�g � f ��x� � g � f �x� � � g��x � 6�
� f � g��x� � f �g�x� � � f �x2 � 5� � ��x2 � 5� � 6 � �x2 � 1 x
0 1 1
3 4�10
�1�5�2
g� f �x��f �g�x��
(c)59. (a)
(b) No, because 2�x � 1� � 2�x � 3� � 1.� f � g��x� � �g � f ��x�
� 2�x � 3� � 1�g � f ��x� � g � f �x� � � g��x � 3�� � �2x � 2� � 2�x � 1�
� f � g��x� � f �g�x� � � f �2x � 1� � ��2x � 1� � 3� x
0 3
0 2 5
1 4 7
�1
g� f �x��f �g�x��
61. (a)
(b) � f
g��2� �f�2�g�2�
�0
2� 0
� f � g��3� � f�3� � g�3� � 2 � 1 � 3 63. (a)
(b) �g � f��2� � g� f �2�� � g�0� � 4
� f � g��2� � f�g�2�� � f�2� � 0
65. Let This is not a unique solution.For example, if and then as well.� f � g��x� � h�x�g�x� � 2x,f�x� � �x � 1�2
f�x� � x2 and g�x� � 2x � 1, then � f � g��x� � h�x�.
67. Let thenThis answer is not unique.
Other possibilities may be:
f�x� � 9�x and g�x� � �x2 � 4�3
f�x� � 3��x and g�x� � 4 � x2 or
f�x� � 3�x � 4 and g�x� � x2 or
� f � g��x� � h�x�.f�x� � 3�x and g�x� � x2 � 4, 69. Let then
Again, this is not a uniquesolution. Other possibilities may be:
or f�x� �1
x � 1 and g�x� � x � 1
f�x� �1
x � 2 and g�x� � x
� f � g��x� � h�x�.f �x� � 1�x and g�x� � x � 2,
71. Let Then (Answer is not unique.)� f � g��x� � h�x�.
f�x� � x2 � 2x and g�x� � x � 4. 73. (a)
(b)
(c) contributes more to at higher speeds.T�x�B�x�
00 60
300
T
B
R
T�x� � R�x� � B�x� �34x �
115x2
75. corresponds to 1995.t � 5
Year 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
140 151.4 162.8 174.2 185.6 197 208.4 219.8 231.2 242.6 254
325.8 342.8 364.4 390.6 421.5 457 497.1 541.8 591.2 645.2 703.8
458.8 475.3 497.9 526.5 561.2 602 648.8 701.7 760.7 825.7 896.8y3
y2
y1
28 Chapter 1 Functions and Their Graphs
87. Let and be the three siblings, in decreasing age. Then and .
(a)
(b) If then and C � 4.B � 8A � 16,
A � 2B � 2�12C � 6� � C � 12
B �12C � 6A � 2BCA, B,
89. Let and be odd functions, and define Then,
since f and g are both odd
Thus, h is even.
Let and be even functions, and define Then,
since f and g are both even
Thus, h is even.
� h�x�.
� f�x�g�x�
h��x� � f��x�g��x�
h�x� � f �x�g�x�.g�x�f �x�
� f�x�g�x� � h�x�.
� ��f�x���g�x�
h��x� � f��x�g��x�
h�x� � f �x�g�x�.g�x�f �x�
28 Chapter 1 Functions and Their Graphs28 Chapter 1 Functions and Their Graphs
81. (a)
represents the number of bacteria as afunction of time.N � T
� 40t2 � 590
� 10�2t � 1�2 � 20�2t � 1� � 600
� N�2t � 1�
�N � T��t� � N�T �t��
83.represents 3 percent of the amount over $500,000.g� f�x�� � g�x � 500,000� � 0.03�x � 500,000� 85. False. but
�g � f��x� � g�x � 1� � 6�x � 1�.� f � g��x� � f�6x� � 6x � 1,
79.
(a)
represents the cost after t hours.
(b) unitsx�4� � 50�4� � 200
C �x�t��
� 3000t � 750
� 60�50t� � 750
C�x�t�� � C �50t�
x�t� � 50t
C�x� � 60x � 750
77. gives the area of the circle as a function of time.
� ��0.6t�2 � 0.36�t2
� A�0.6t�
�A � r��t� � A�r�t��
�A � r��t�
(c)
, or 4 hours 45 minutest � 4.75
30,000
103,0000
(b)
At time there are 2030 bacteria.
(c) when hours.t 2.3N � 800
t � 6,
�N � T��6� � 10�132� � 20�13� � 600 � 2030
Section 1.6 Inverse Functions 29
Section 1.6 Inverse Functions
■ Two functions f and g are inverses of each other if for every x in the domain of g andfor every x in the domain of f.
■ Be able to find the inverse of a function, if it exists.
1. Replace with y.
2. Interchange x and y.
3. Solve for y. If this equation represents y as a function of x, then you have found If this equationdoes not represent y as a function of x, then f does not have an inverse function.
■ A function f has an inverse function if and only if no horizontal line crosses the graph of f at more thanone point.
■ A function f has an inverse function if and only if f is one-to-one.
f�1�x�.
f�x�
g� f�x�� � xf�g�x�� � x
1.
f�1� f �x�� � f�1�6x� �16�6x� � x
f � f�1�x�� � f �16 x� � 6�1
6 x� � x
f�1�x� �16 x
f �x� � 6x
Vocabulary Check
1. inverse, 2. range, domain 3.
4. one-to-one 5. Horizontal
y � x f �1
91. which shows that g is even.
which shows that h is odd.
� �12 � f�x� � f��x�� � �h�x�,
h��x� �12 � f��x� � f����x��� �
12 � f ��x� � f �x��
g��x� �12 � f��x� � f����x��� �
12 � f��x� � f �x�� � g�x�,
93. �other answers possible��0, �5�, �1, �5�, �2, �7� 95.
�other answers possible���24, 0�, ���24, 0�, �0, �24�
97.
y � 10x � 38 � 0
y � 2 � 10�x � 4�
y � ��2� �8 � ��2�
�3 � ��4��x � ��4�� 99.
30x � 11y � 34 � 0
11y � 11 � �30x � 45
y � 1 �5
�11�6�x �32� � �
3011�x �
32�
y � ��1� �4 � ��1�
��1�3� � �3�2��x �32�
3.
f�1� f �x�� � f�1�x � 7� � �x � 7� � 7 � x
f � f�1�x�� � f �x � 7� � �x � 7� � 7 � x
f�1�x� � x � 7
f �x� � x � 7
30 Chapter 1 Functions and Their Graphs
11. (a)
(b)
Note that the entries in the tables are the same except that the rows are interchanged.
g � f �x� � � g �x3 � 5� � 3��x3 � 5� � 5 � 3�x3 � x
f �g�x� � � f � 3�x � 5� � � 3�x � 5�3� 5 � �x � 5� � 5 � x
x 0 1
4 5 6�3�22f �x�
�1�2�3
x 4 5 6
0 1�1�2�3g�x�
�3�22
13. (a)
(b)
Note that the entries in the tables are the same except that the rows are interchanged.
g � f �x�� � g���x � 8� � 8 � ���x � 8�2� 8 � �x � 8� � x
�Since x ≤ 0, �x2 � �x�
x ≤ 0f �g�x� � � f �8 � x2� � ���8 � x2� � 8 � ��x2 � ���x� � x
x 8 9 12 17 24
0 �4�3�2�1f �x�
x 0
8 9 12 17 24g�x�
�4�3�2�1
9. (a)
(b)
Note that the entries in the tables are the same except that the rows are interchanged.
g� f �x�� � g��72
x � 3� � �2��7
2x � 3� � 67
� ��7x � 6 � 6
7�
7x7
� x
f �g�x� � � f ��2x � 67 � � �
72��
2x � 67 � � 3 �
2x � 62
� 3 � �x � 3� � 3 � x
x 2 0
4 11 18�3�10f �x�
�6�4�2
x 4 11 18
2 0 �6�4�2g�x�
�3�10
5.
f �1� f �x�� � f �1�2x � 1� ��2x � 1� � 1
2�
2x2
� x
f � f �1�x�� � f �x � 12 � � 2�x � 1
2 � � 1 � �x � 1� � 1 � x
f �1�x� �x � 1
2
7.
f�1� f�x�� � f �1� 3�x � � � 3�x�3� x
f � f�1�x�� � f�x3� � 3�x3 � x
f�1�x� � x3
Section 1.6 Inverse Functions 31
19.
g � f�x�� � g�1 � x3�� 3�1 � �1 � x3� � 3�x3 � x
f�g�x�� � f � 3�1 � x � � 1 � � 3�1 � x ��3� 1 � �1 � x� � x
Reflections in the line y � x
−4
−6 6
4
f
g
21. The inverse is a line through
Matches graph (c).
��1, 0�.
15.
Reflections in the line y � x
−4
−6 6
4
f g
g� f�x�� � g�x3� � 3�x3 � x
f�g�x�� � f � 3�x � � � 3�x �3� x 17.
Reflections in the line y � x
00 15
10
f
g
� ��x � 4 �2� 4 � x
g� f�x�� � g��x � 4 � � ��x2 � 4� � 4 � x
f�g�x�� � f�x2 � 4�, x ≥ 0
23. The inverse is half a parabola starting at
Matches graph (a).
�1, 0�.
25.
(a)
Reflection in the line y � x
−4
−6 6
4
fg
g�x� �x2
f �x� � 2x,
(b)
The entries in the tables are the same, except thatthe rows are interchanged.
x 0 1 2
0 2 4�2�4f �x�
�1�2
x 0 2 4
0 1 2�1�2g�x�
�2�4
27.
(a)
Reflection in the line y � x
−10
−12 12
6
f f
gg
g�x� � �5x � 1x � 1
�5x � 11 � x
f �x� �x � 1x � 5,
(b)
The entries in the tables are the same, except thatthe rows are interchanged.
x 0 3 525
14�
15�
12�1f �x�
�1�2
x
0 3 5�1�2g�x�
25
14�
15�
12�1
32 Chapter 1 Functions and Their Graphs
47.
is not one-to-one.
This does not represent y as a function of x. f doesnot have an inverse.
f
y � ± 4�x
x � y4
y � x4
f�x� � x4
49.
is one-to-one and has an inverse.f
f �1�x� �5x � 4
3
5x � 4
3� y
5x � 4 � 3y
5x � 3y � 4
x �3y � 4
5
y �3x � 4
5
−9
−6
9
6
f �x� �3x � 4
5
51. is not one-to-one, and does not have an
inverse. For example, f �1� � f ��1� � 1.
f �x� �1
x2
32 Chapter 1 Functions and Their Graphs
39.
is not one-to-one becausesome horizontal lines intersectthe graph twice.
−2
−6 6
6
h
h�x� � �16 � x2
41.
is not one-to-one because thehorizontal line intersectsthe graph at every point on thegraph.
−2
−12 12
14
y � 10f
f �x� � 10 43.
is one-to-one because a horizontal line willintersect the graph at mostonce.
−4
−10 2
4
g
g�x� � �x � 5�3 45.
is not one-to-one becausesome horizontal lines intersectthe graph more than once.
−8
−12 12
8
h
h�x� � x � 4 � x � 4
33. It is the graph of a one-to-onefunction.
35.
is one-to-one because a horizontal line will intersect the graph at most once.
−2
−4 8
6
f
f �x� � 3 �12
x 37.
is not one-to-one becausesome horizontal lines intersectthe graph twice.
−1
−3 3
3
h
h�x� �x2
x2 � 1
29. Not a function 31. It is the graph of a one-to-onefunction.
Section 1.6 Inverse Functions 33
57.
since
x ≥ 0f�1�x� � �x � 2,
y ≤ 2x ≥ 0,y � �x � 2,
x � �y � 2
y � 2 ≤ 0.x � �� y � 2�
y ≤ 2, x ≥ 0x � y � 2, y � x � 2
−9
−6
9
6 f �x� � x � 2, x ≤ 2, y ≥ 0
59.
f�1�x� �x � 3
2
y �x � 3
2
x � 2y � 3
y � 2x � 3
f�x� � 2x � 3 61.
f�1�x� � 5�x
y � 5�x
x � y5
y � x5
f�x� � x5
Reflections in the line y � x
−4
−6 6
4
ff−1
55.
is one to one.
This is a function of x,so f has an inverse.
f�1�x� �x2 � 3
2, x ≥ 0
−3
−2
9
6f
y �x2 � 3
2, x ≥ 0, y ≥ �
3
2
x2 � 2y � 3, x ≥ 0, y ≥ �3
2
x � �2y � 3, y ≥ �3
2, x ≥ 0
y � �2x � 3, x ≥ �3
2, y ≥ 0
f�x� � �2x � 3 ⇒ x ≥ �3
2, y ≥ 0
Reflections in the line y � x
−2
−4 8
6
ff−1
53.
is one-to-one.
This is a function of x,so f has an inverse.
f�1�x� � �x � 3, x ≥ 0
−3
−3
9
5f
y � �x � 3, x ≥ 0, y ≥ �3
�x � y � 3, y ≥ �3, x ≥ 0
x � � y � 3�2, y ≥ �3, x ≥ 0
y � �x � 3�2, x ≥ �3, y ≥ 0
f �x� � �x � 3�2, x ≥ �3, y ≥ 0
65.
f �1�x� � �4 � x2, 0 ≤ x ≤ 2
y � �4 � x2
y2 � 4 � x2
x2 � 4 � y2
x � �4 � y2
y � �4 � x2
f �x� � �4 � x2, 0 ≤ x ≤ 263.
f�1�x� � x5�3
y � x5�3
x � y3�5
y � x3�5
f�x� � x3�5
Reflections in the line y � x
−2
−3 3
2
f
f−1
Reflections in the line y � x
00 4
3
f = f−1
79.
–4 –3 1 2 3
–3
–2
–1
1
2
3
4
x
y
x
1 2
3 3
�2�1
�4�2
f �x� x
2 1
3 3
�1�2
�2�4
f�1�x�
34 Chapter 1 Functions and Their Graphs
75. Let
Domain Range
Domain Range f �1: y ≥ 0f �1: x ≤ 5
f : y ≤ 5f : x ≥ 0
f �1�x� ��5 � x2
y � ��5 � x��2
y2 �x � 5�2
�5 � x
2
x � 5 � �2y2
x � �2y2 � 5
y � �2x2 � 5
f�x� � �2x2 � 5, x ≥ 0. 77. Let and
because
Domain Range
Domain Range f �1: y ≥ 4f �1: x ≥ 1
f : y ≥ 1f : x ≥ 4
f �1�x� � x � 3, x ≥ 1
y � x � 3
x � y � 3
x ≥ 4. y � x � 3
y � x � 4 � 1
y ≥ 1.f �x� � x � 4 � 1, x ≥ 4
71. If we let then f has an inverse. Note: We could also let
Thus, f�1�x� � x � 2, x ≥ 0.
x � 2 � y, x ≥ 0, y ≥ �2
x � y � 2, x ≥ 0, y ≥ �2
y � x � 2, x ≥ �2, y ≥ 0
f�x� � x � 2 when x ≥ �2
f�x� � x � 2, x ≥ �2
x ≤ �2.��f�x� � x � 2, x ≥ �2, 73. Let
Domain Range
Domain Range f �1: y ≥ �3f �1: x ≥ 0
f : y ≥ 0f : x ≥ �3
f �1�x� � �x � 3
y � �x � 3
�x � y � 3
x � � y � 3�2
y � �x � 3�2
f �x� � �x � 3�2, x ≥ �3.
34 Chapter 1 Functions and Their Graphs
67.
Reflections in the line y � x
f �1�x� �4x
y �4x
xy � 4
x �4y
y �4x
−4
−6 6
4
f = f−1
f �x� �4x
69. If we let then f has an inverse. Note: We could also let
Thus, f�1�x� � �x � 2, x ≥ 0.
�x � 2 � y, x ≥ 0, y ≥ 2
�x � y � 2, x ≥ 0, y ≥ 2
x � �y � 2�2, x ≥ 0, y ≥ 2
y � �x � 2�2, x ≥ 2, y ≥ 0
f �x� � �x � 2�2, x ≥ 2, y ≥ 0
x ≤ 2.��f�x� � �x � 2�2, x ≥ 2,
Section 1.6 Inverse Functions 35
81. because f �12� � 0.f �1�0� �
12 83. � f � g��2� � f �3� � �2
85. f �1�g�0�� � f �1�2� � 0 87. �g � f �1 ��2� � g �0� � 2
89.
The graph of the inverserelation is an inversefunction since it satisfiesthe Vertical Line Test. −3
−6 6
5
f−1
ff�x� � x3 � x � 1
97. Now find the inverse of
Note: � f � g��1 � g�1� f �1
� f � g��1�x� � 2 3�x � 3
3�8�x � 3� � y
8�x � 3� � y3
x � 3 �18 y3
x �18 y3 � 3
y �18 x3 � 3
� f � g��x� �18 x3 � 3:� fg��x� � f�g�x�� � f�x3� �
18x3 � 3
91.
The graph of the inverserelation is not an inversefunction since it does notsatisfy the Vertical Line Test.
−4
−6 6
4
g−1
gg�x� �3x2
x2 � 1
In Exercises 93–97,
93.
95. � f �1� f�1��6� � f�1� f�1 �6�� � f�1�8�6 � 3�� � f�1�72� � 8�72 � 3� � 600
� f�1� g�1��1� � f�1�g�1�1�� � f�1� 3�1 � � 8� 3�1 � 3� � 8�1 � 3� � 32
fx� � 18x � 3, f�1x� � 8x � 3�, gx� � x3, g�1x� � 3�x.
In Exercises 99 and 101, fx� � x � 4, f�1x� � x � 4, gx� � 2x � 5, g�1x� �x � 5
2.
99.
�x � 1
2
��x � 4� � 5
2
� g�1�x � 4�
�g�1� f�1��x� � g�1� f�1�x��
101. Now find the inverse of
Note that � f � g��1�x� � �g�1� f�1��x�; see Exercise 99.
� f � g��1�x� �x � 1
2
y �x � 1
2
x � 1 � 2y
x � 2y � 1
y � 2x � 1
� f � g��x� � 2x � 1:� f � g��x� � f�g�x�� � f�2x � 5� � �2x � 5� � 4 � 2x � 1.
36 Chapter 1 Functions and Their Graphs
Section 1.7 Linear Models and Scatter Plots
■ You should know how to construct a scatter plot for a set of data
■ You should recognize if a set of data has a positive correlation, negative correlation, or neither.
■ You should be able to fit a line to data using the point-slope formula.
■ You should be able to use the regression feature of a graphing utility to find a linear model for a set of data.
■ You should be able to find and interpret the correlation coefficient of a linear model.
Vocabulary Check
1. positive 2. negative 3. fitting a line to data 4. �1, 1
109. We will show that for all x in their domains.Let then Hence,Thus, g�1
� f�1 � � f � g��1. � y � � f � g��1�x�. � g�1�g�y���g�1
� f�1��x� � g�1� f�1�x��f �g� y�� � x ⇒ f�1�x� � g� y�.y � � f � g��1�x� ⇒ � f � g�� y� � x
� f � g��1�x� � �g�1� f�1��x�
111. No, the graphs are not reflections of each other in the line y � x.
113. Yes, the graphs are reflections of each other in theline y � x.
115. Yes. The inverse would give the time it took tocomplete n miles.
107. False. is even, but does not exist.f �1f �x� � x2
117. No. The function oscillates.
119.27x3
3x2 � 9x, x � 0 121.x2 � 366 � x
��x � 6��x � 6�
��x � 6� �x � 6�1
� �x � 6, x � 6
123.
Yes, y is a function of x.
y � 4x � 3
4x � y � 3 125.
No, y is not a function of x.
y � ±�9 � x2
x2 � y2 � 9 127.
Yes, y is a function of x.
y � �x � 2
103. (a) Yes, f is one-to-one. For each European shoesize, there is exactly one U.S. shoe size.
(b)
(c) because
(d)
(e) f �1� f �13�� � f �1 �47� � 13
f � f �1 �41�� � f �8� � 41
f �10� � 43.f �1 �43� � 10
f �11� � 45
105. (a) Yes, is one-to-one, so exists.
(b) gives the year corresponding to the 10 values in the second column.
(c) because
(d) No, because f �11� � f �15� � 690.4.
f �10� � 650.3.f �1�650.3� � 10
f �1
f �1f
Section 1.7 Linear Models and Scatter Plots 37
11. (a)
(b)
(c) or
(d) If cm.d � 0.066�55� � 3.63F � 55,
F � 15.13d � 0.096d � 0.066F
d � 0.07F � 0.3
Force
Elo
ngat
ion
d
F20 40 60 80 100
1
2
3
4
5
6
7
13. (a)
(b)
(c)
Yes, the model is a good fit.
(d) For 2005, and or $1,516,500.
For 2010, and or $2,197,000.
Yes, the answers seem reasonable.
(e) The slope is 136.1. It says that the mean salaryincreases by $136,100 per year.
y � 2197,t � 10
y � 1516.5,t � 5
−1 50
1600
y � 136.1t � 836
−1 50
1600
9. (a)
(b)
Correlation coefficient: 0.90978
(c)
(d) Yes, the model appears valid.
−4
−1
8
7
y � 0.95x � 0.92
y
x
(0, 2)
(1, 1)
(2, 2)
(5, 6)
(3, 4)
−1−2 1 2 3 4 5 6
1
2
3
4
5
6
y = x −32
12
1. (a) (b) Yes, the data appears somewhat linear.The more experience, corresponds tohigher sales, y.
x,
Years of experience
Mon
thly
sal
es(i
n th
ousa
nds
of d
olla
rs)
y
x1 2 3 4
10
20
30
40
50
60
3. Negative correlation— decreases as increases.xy 5. No correlation
7. (a)
(b)
Correlation coefficient: 0.95095
(c)
(d) Yes, the model appears valid.
−4
−1
5
5
y � 0.46x � 1.62
y
x−1−2−4 1 2 3 4
−1
−2
−3
1
2
3
4
5
(−3, 0)(−1, 1)
(0, 2)(2, 3)
(4, 3)
y = x +23
53
21. True. To have positive correlation, the y-values tendto increase as x increases.
38 Chapter 1 Functions and Their Graphs
17. (a)
(b)
(c)
The model is not a good fit.
(d) For 2050, and or 542,000people. Answers will vary.
P � 542,t � 50
0 350
700
P � 0.6t � 512
0 350
70015. (a)
(b)
Correlation coefficient: 0.99544
(c)
(d) The model is a good fit.
(e) For 2005,
For 2010,
(f) Answers will vary.
y1 � $46.74.t � 20,
y1 � $38.98.t � 15,
−1 150
60
C � 1.552t � 15.70
−1 150
60
23. Answers will vary.
25.
(a)
(b)
� 2w2 � 5w � 7
f�w � 2� � 2�w � 2�2 � 3�w � 2� � 5
f��1� � 2 � 3 � 5 � 10
f�x� � 2x2 � 3x � 5 27.
(a) (b) h�0� � 1 � 0 � 1h�1� � 2�1� � 3 � 5
h�x� � �1 � x2,2x � 3,
x ≤ 0 x > 0
29.
x � �915 � �
35
15x � �9
6x � 1 � �9x � 8 31.
x � �14, 32
�4x � 1��2x � 3� � 0
8x2 � 10x � 3 � 0 33.
�7 ± �17
4
x �7 ± �49 � 4�4��2�
4
2x2 � 7x � 4 � 0
19. (a)
Correlation coefficient: 0.79495
(b)
5 180
2000
T � 36.7t � 926 (c) The slope indicated the number of new stores opened per year.
(d)
The number of stores will exceed 1800 near the end of 2013.
t > 23.8
36.7t > 874
T � 36.7t � 926 > 1800
(e)
The model is not a good fit, especially around t � 14.
Year 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006
Data 1130 1182 1243 1307 1381 1475 1553 1308 1400 1505
Model 1183 1220 1256 1293 1330 1366 1403 1440 1477 1513
Review Exercises for Chapter 1
Review Exercises for Chapter 1 39
1.
−1−4−5 1 2 4 5
−4
−5
2
4
5
x
y 3.
–4 –2 2 4 6 8
–4
–2
4
6
8
x
(−3, 2) (8, 2)
y
m �2 � 2
8 � ��3� �011
� 0 5.
5, 52
2 4 6
−2
2
4
6
, 132(
((
(
x
y
m ��5�2� � 1
5 � �3�2��
3�2
7�2�
3
7
7.
–6 –4 –2 2 4 6
–4
–2
2
6
8
x
(−4.5, 6)
(2.1, 3)
y
m �3 � 6
2.1 � ��4.5��
�3
6.6� �
30
66� �
5
11
��4.5, 6�, �2.1, 3� 9. (a)
(b) Three additional points:
(other answers possible)
�10 � 4, 1 � 1� � �14, 2�
�6 � 4, 0 � 1� � �10, 1�
�2 � 4, �1 � 1� � �6, 0�
�x � 4y � 6 � 0
4y � 4 � x � 2
y � 1 �14
�x � 2�
11. (a)
(b) Three additional points:
(other answers possible)
�4 � 2, 1 � 3� � �6, 4�
�2 � 2, �2 � 3� � �4, 1�
�0 � 2, �5 � 3� � �2, �2�
�3x � 2y � 10 � 0
2y � 10 � 3x
y � 5 �32�x � 0� 13. (a)
(b) Three additional points:
(other answers possible)
�115 � 1, �7 � 1� � �16
5 , �8��6
5 � 1, �6 � 1� � �115 , �7�
�15 � 1, �5 � 1� � �6
5, �6�
5x � 5y � 24 � 0
5y � 25 � �5x � 1
y � 5 � �x �15
y � 5 � �1�x �15�
15. (a)
(b) Three additional points:
(other answers possible)
�0, 6�, �1, 6�, �2, 6�
y � 6 � 0
y � 6 � 0�x � 2� 17. (a) is undefined means that the line is vertical.
(b) Three additional points:
(other answers possible)
�10, 0�, �10, 1�, �10, 2�
x � 10 � 0
m
40 Chapter 1 Functions and Their Graphs
29.
(a) Parallel slope:
(b) Perpendicular slope:
−4
−6
8
2
y � �45 x �
25
4x � 5y � 2 � 0
5y � 10 � �4x � 12
y � ��2� � �45�x � 3�
m � �45
y �54x �
234
0 � 5x � 4y � 23
4y � 8 � 5x � 15
y � ��2� �54�x � 3�
m �54
5x � 4y � 8 ⇒ y �54x � 2 and m �
54 31. is a vertical line; the slope is not defined.
(a) Parallel line:
(b) Perpendicular slope:
Perpendicular line:
−7
−3
2
3
� 0 ⇒ y � 2
y � 2 � 0�x � 6�
m � 0
x � �6
x � 4
40 Chapter 1 Functions and Their Graphs
19.
−3
−3 3
1
�Slope � 0�
� 0�x � 2� � 0 ⇒ y � �1
y � 1 ��1 � 14 � 2
�x � 2� 21.
−4
−6 6
4
�27
�x � 1� �27
x �27
⇒ y �27
x �27
y � 0 �2 � 0
6 � ��1� �x � 1�
23. corresponds to 2008.
Point: , slope:
V � 850t � 5700
V � 12,500 � 850�t � 8�
850�8, 12,500�
t � 8 25.
Point:
V � 42.70t � 283.90
V � 625.50 � 42.70�t � 8�
�8, 625.50�
m � 42.70
27.
For the fourth quarter let Then we have S � 25,000�4� � 110,000 � $210,000.t � 4.
S � 25,000t � 110,000
S � 160,000 � 25,000�t � 2�
m �185,000 � 160,000
3 � 2� 25,000
�2, 160,000�, �3, 185,000�
Review Exercises for Chapter 1 41
33. (a) Not a function. 20 is assignedtwo different values.
(b) Function
(c) Function
(d) Not a function. No value isassigned to 30.
35. No, y is not a function of x. Some x-values correspond to two y-values. For example,
corresponds to and y � �4.
y � 4x � 1
37.
Each x value,corresponds to only one y-value so y is a function of x.
x ≤ 1,
y � �1 � x
39.
(a)
(b)
(c)
(d) f �x � 1� � �x � 1�2 � 1 � x2 � 2x � 2
f �b3� � �b3�2 � 1 � b6 � 1
f ��3� � ��3�2 � 1 � 10
f �1� � 12 � 1 � 2
f �x� � x2 � 1 41.
(a)
(b)
(c)
(d) h�2� � 22 � 2 � 6
h�0� � 02 � 2 � 2
h��1� � 2��1� � 1 � �1
h��2� � 2��2� � 1 � �3
h�x� � �2x � 1,x2 � 2,
x ≤ �1x > �1
43. The domain of is all real numbers
x � �2.
f �x� �x � 1x � 2
45.
Domain:
Domain: ��5, 5�
�5 � x��5 � x� ≥ 0
25 � x2 ≥ 0
f �x� � �25 � x2
47. The domain of is all real numbers
s � 3.
g�5� �5s � 53s � 9
49. (a)
(b)
� 2.85x � 16,000
� 8.20x � �16,000 � 5.35x�
P�x� � R�x� � C�x�
C�x� � 16,000 � 5.35x
51.
� 4x � 2h � 3, h � 0
�4xh � 2h2 � 3h
h
f �x � h� � f �x�
h�
�2x2 � 4xh � 2h2 � 3x � 3h � 1� � �2x2 � 3x � 1�h
� 2x2 � 4xh � 2h2 � 3x � 3h � 1
f �x � h� � 2�x � h�2 � 3�x � h� � 1
f �x� � 2x2 � 3x � 1
53. Domain: All real numbers
Range:
−4
−6 6
4
y ≤ 3
55. Domain:
Range:
−4
−9 9
8
0 ≤ y ≤ 6
36 � x2 ≥ 0 ⇒ x2 ≤ 36 ⇒ �6 ≤ x ≤ 6
73.
Even
� f �x�
� x2 � 6
f ��x� � ��x�2 � 6
79. f �x� � �2 is a constant function.
75.
f is even.
� f �x�
� �x2 � 8�2
f ��x� � ���x�2 � 8�2 77. and
Neither even nor odd
(Note that the domain of is x ≥ 0.)
f
f ��x� � �f �x�f ��x� � 3��x�5�2 � f �x�
81. is the parent function. is obtained from by a horizontal shift two units to the right,followed by a vertical shift one unit upward.
f �x� � �x � 2�2 � 1 � g�x � 2� � 1
gfg�x� � x2
83. is obtained from by a vertical shift three units upward.
g�x� � f �x� � 3
f �x� � �x�g�x� � �x� � 3
42 Chapter 1 Functions and Their Graphs
61.
(a)
(b) Increasing on and
Decreasing on ��1, 1�
�1, �����, �1�
−6
−9 9
6
f �x� � x3 � 3x
63.
(a)
(b) Increasing on �6, ��
00 21
14
f �x� � x�x � 6 65.
Relative minima: and
Relative maximum:
−18
−4
18
20
�0, 16�
�2, 0���2, 0�
f �x� � �x2 � 4�2
67.
Relative maximum: �3, 27�
−10
−10
10
30
h �x� � 4x3 � x4 69.
1
56
1 2 4 5 6−1−3−4−5−6−2−3−4−5−6
x
y
f �x� � �3x � 5,x � 4,
x < 0 x ≥ 0
71.
–5 –1–2 1 2 3 4
–3
–2
3
4
5
6
x
y
f �x� � x � 3
42 Chapter 1 Functions and Their Graphs
57. (a)
(b) is a function of
−6
−9 9
6
x.y
y �x2 � 3x
659. (a)
(b) is not a function of
−6
−9 9
6
x.y
y � ±�2 � 3x
y2 � 2 � 3x
3x � y2 � 2
Review Exercises for Chapter 1 43
87.
is a vertical shifttwo units downward.y � f �x� � 2
−4−6
4
2
6 8 10
−6
−8
−10
−12
−4
x
y
(−4, −6)
(−1, 0) (4, 0)
(8, −6)
89. (a)
(b) is a vertical shift sixunits downward.
(c)
(d) h�x� � f �x� � 6
1
2
3
−2
−3
−7
−1−3−4−5 1 3 4 5x
y
h
f �x� � x2
93. (a)
(b) is a horizontal shift two units to the right, a reflection in the axis, followed by a vertical shift eight units downward.
(c)
(d) h�x� � �f �x � 2� � 8
−2−4−6−8 2
2
4 6 8
−6
−8
−10
−12
−14
−4
−2
x
y
x-h
f �x� � x291.
(a)
(b) The graph of h is a horizontal shift of f twounits to the right, followed by a vertical shiftfive units upward.
(c)
(d) � f �x � 2� � 5h �x� � �x � 2�3 � 5
–2 –1
–2–3
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
x
y
f �x� � x3
h �x� � �x � 2�3 � 5
85.
is a reflection in the y-axis.y � f ��x�
−2−2−4−8−10 2
4
6
8
4 6
−6
−8
−4
x
y
(−8, −4)
(−4, 2) (1, 2)
(4, −4)
95.
(a)
(b) The graph of h is a reflection of f in the x-axis,followed by a vertical shift five units upward.
(c)
(d)
� �f �x� � 5
h�x� � ��x � 5
1
32
4
78
65
y
–1
–2
54321 6 7 8 9x
f �x� � �x
h�x� � ��x � 5 97.
(a)
(b) The graph of is a horizontal shift of one unitto the right, followed by a vertical shift threeunits upward.
(c)
(d) h�x� � f �x � 1� � 3
−1−2−3−4 1
1
2
3
4
5
6
2 3 4−1
−2
x
y
h
f �x� � �x
h�x� � �x � 1 � 3
44 Chapter 1 Functions and Their Graphs
119.
f �1� f �x�� � f �1�6x� �16�6x� � x
f � f �1�x�� � f �16 x� � 6�1
6 x� � x
f �1�x� �16 x
f �x� � 6x 121.
� 2�12
x � 3 � 3� � 2�12
x� � x
f �1� f �x�� � f �1�12
x � 3�
�12
�2�x � 3�� � 3 � x � 3 � 3 � x
f � f �1�x�� � f �2�x � 3��
f �x� �12
x � 3 ⇒ f �1�x� � 2�x � 3� � 2x � 6
44 Chapter 1 Functions and Their Graphs44 Chapter 1 Functions and Their Graphs44 Chapter 1 Functions and Their Graphs
101.
� �5 � 2 � �7
� �3 � 2�4�� � �4
� f � g��4� � f�4� � g�4� 103.
� �42
� �47 � 5
� f � g��25� � f �25� � g�25�
105.
� �1��5� � 5
� �3 � 2�1���3�1�2 � 2�
� fh��1� � f�1�h�1� 107.
� 23
� 3��7 �2� 2
� h��7 ��h � g��7� � h�g�7�� 109.
� �97
� f �50�
� f � h���4� � f �h��4��
111.
� �x � 3�2 � h�x�
� f � g��x� � f �x � 3�
f �x� � x2, g�x� � x � 3 113.
� f � g��x� � f �4x � 2� � �4x � 2 � h�x�
f �x� � �x, g�x� � 4x � 2
115.
� f � g��x� � f �x � 2� �4
x � 2� h�x�
f �x� �4x, g�x� � x � 2 117. 3
140
0
y1
y2
y1 + y2
99.
(a)
(b) is a vertical shrink, followed by a reflectionin the axis, followed by a vertical shift nineunits upward.
x-h
f �x� � �x�h�x� � �
12�x� � 9
(c)
(d) h�x� � �12 f �x� � 9
−2−4−6−8 2
2
4
6
10
12
4 6 8
−4
−2
x
y
Review Exercises for Chapter 1 45
123. (a)
Reflection in the line y � x
−6
−9 9
6
g
f
(b)
The entries in the table are the same exceptthat their rows are interchanged.
x 23 7 3
0 1 3�1�5g�x�
�9�1
x 0 1 3
23 7 3 �9�1f �x�
�1�5
125.
passes the Horizontal Line Test,and hence is one-to-one and has an inverse� f�1�x� � 2�x � 3��.
f �x� �12 x � 3
−6
−9 9
6 127.
passes the Horizontal Line Test, and
hence is one-to-one.
h�t� �2
t � 3
−6
−9 9
6
133.
f �1�x� � x2 � 10 , x ≥ 0
x2 � 10 � y
x2 � y � 10
x � �y � 10 , y ≥ �10, x ≥ 0
y � �x � 10, x ≥ �10, y ≥ 0
f �x� � �x � 10
129.
f�1�x� � 2x � 10
y � 2�x � 5�
x � 5 �12
y
x �12
y � 5
y �12
x � 5 131.
f �1�x� � 3�x � 34
x � 3
4� y3
x � 3 � 4y3
x � 4y3 � 3
y � 4x3 � 3
f �x� � 4x3 � 3
135. Negative correlation
137. (a)
Exam score
Gra
de-p
oint
ave
rage
y
x65 70 75 80 85 90 95
1
2
3
4
(b) Yes, the relationship is approximately linear.Higher entrance exam scores, are associatedwith higher grade-point averages, y.
x,
46 Chapter 1 Functions and Their Graphs46 Chapter 1 Functions and Their Graphs46 Chapter 1 Functions and Their Graphs46 Chapter 1 Functions and Their Graphs
(b) (Approximations will vary.)
(c)
(d) For m/sec.t � 2.5, S 24.7
s � 9.7t � 0.4; 0.99933
s 10t
141. y � 95.174x � 458.423 143. The model does not fit well.
145. False. and g��1� � �52 � 28g�x� � ���x � 6�2 � 3� � ��x � 6�2 � 3
147. False. or satisfies f � f �1.f �x� � x f �x� �1x
139. (a)
Time (in seconds)
Spee
d (i
n m
eter
s pe
r se
cond
)
t1 2 3 4
5
10
15
20
25
30
35
40
s
Chapter 1 Practice Test
Practice Test for Chapter 1 47
1. Find the slope of the line passing through the points and �1, 3�.��2, 2�
10. Determine the open interval(s) on which the function is increasing.f �x� � 12x � x3
2. Find an equation for the line passing through the points Use a graphing utility to sketch agraph of the line.
�3, �2� and �4, �5�.
3. Find an equation of the line that passes through the point and has slope Use a graphing utility tosketch a graph of the line.
�3.��1, 5�
4. Find the slope-intercept form of the line that passes through the point and is perpendicularto 3x � 5y � 7.
��3, 2�
5. Does the equation represent y as a function of x?x4 � y4 � 16
6. Evaluate the function at the points x � 0, x � 2, and x � 4.f�x� � �x � 2���x � 2�
9. Use a graphing utility to sketch the graph of the function and determine if the function is even, odd,or neither.
f�x� � 3 � x6
11. Use a graphing utility to approximate any relative minimum or maximum values of the function y � 4 � x � x3.
7. Find the domain of the function f�x� � 5��x2 � 16�.
8. Find the domain of the function g�t� � �4 � t.
12. Compare the graph of with the graph of y � x3.f�x� � x3 � 3
13. Compare the graph of with the graph of y � �x.f �x� � �x � 6
14. Find if and What is the domain of g � f ?g�x� � x2 � 2.f�x� � �xg � f
15. Find if and What is the domain of f�g?g�x� � 16 � x4.f �x� � 3x2f�g
16. Show that and are inverse functions algebraically and graphically.g�x� �x � 1
3f�x� � 3x � 1
17. Find the inverse of Graph f and in the same viewing rectangle.f�1f �x� � �9 � x2, 0 ≤ x ≤ 3.
18. Use a graphing utility to find the least squares regression line for the points Graphthe points and the line.
�4, 5�.�3, 3�,�0, 1�,��1, 0�,