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68
CHAPTER 2
Functions and Their Graphs
2.1 Assess Your Understanding
‘Are You Prepared?’
Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
x - 3
1. The inequality -1 6 x 6 3 can be written in interval nota3. The domain of the variable in the expression
tion as _____. (pp. 1020–1022)
x + 4
is _____. (pp. 953–954)
1
2. If x = -2, the value of the expression 3x2 - 5x +
4. Solve the inequality: 3 - 2x 7 5. Graph the solution set.
x
(pp. 1024–1025)
is _____. (p. 953)
Concepts and Vocabulary
5. If f is a function defined by the equation y = f1x2, then x is
called the _____ variable and y is the _____ variable.
6. The set of all images of the elements in the domain of a
function is called the _____.
7. If the domain of f is all real numbers in the interval 30, 74
and the domain of g is all real numbers in the interval
3-2, 54, the domain of f + g is all real numbers in the interval _____.
f
consists of numbers x for which g1x2 _____ 0
g
that are in the domains of both _____ and _____.
8. The domain of
Skill Building
9. If f1x2 = x + 1 and g1x2 = x3,
then _________ = x3 - 1x + 12.
10. True or False: Every relation is a function.
11. True or False: The domain of 1f # g21x2 consists of the
numbers x that are in the domains of both f and g.
12. True or False: The independent variable is sometimes
referred to as the argument of the function.
13. True or False: If no domain is specified for a function f, then
the domain of f is taken to be the set of real numbers.
x2 - 4
14. True or False: The domain of the function f1x2 =
is
x
5x ƒ x Z ;26.
In Problems 15–26, determine whether each relation represents a function. For each function, state the domain and range.
15.
16.
Person
Birthday
Father
Daughter
Elvis
Jan. 8
Bob
Kaleigh
Mar. 15
John
Linda
Marissa
Sept. 17
Chuck
Marcia
Colleen
17. Hours Worked
20 Hours
Diane
18.
Salary
$200
$300
19.
21.
23.
25.
Beth
30 Hours
$350
40 Hours
$425
512, 62, 1-3, 62, 14, 92, 12, 1026
511, 32, 12, 32, 13, 32, 14, 326
51-2, 42, 1-2, 62, 10, 32, 13, 726
51-2, 42, 1-1, 12, 10, 02, 11, 126
20.
22.
24.
26.
Level of Education
Average Income
Less than 9th grade
9th-12th grade
High School Graduate
Some College
College Graduate
$18,120
$23,251
$36,055
$45,810
$67,165
51-2, 52, 1-1, 32, 13, 72, 14, 1226
510, -22, 11, 32, 12, 32, 13, 726
51-4, 42, 1-3, 32, 1-2, 22, 1-1, 12, 1-4, 026
51-2, 162, 1-1, 42, 10, 32, 11, 426
In Problems 27–38, determine whether the equation defines y as a function of x.
1
x
27. y = x2
28. y = x3
29. y =
31. y2 = 4 - x2
32. y = ; 21 - 2x
33. x = y2
30. y = ƒ x ƒ
34. x + y2 = 1
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
SECTION 2.1
3x - 1
37. 2x2 + 3y2 = 1
x + 2
In Problems 39–46, find the following values for each function:
(a) f102
(b) f112 (c) f1-12
(d) f1-x2
(e) -f1x2
(f) f1x + 12
x
39. f1x2 = 3x2 + 2x - 4
40. f1x2 = -2x2 + x - 1
41. f1x2 = 2
x + 1
2x + 1
43. f1x2 = ƒ x ƒ + 4
44. f1x2 = 3x2 + x
45. f1x2 =
3x - 5
35. y = 2x2 - 3x + 4
Functions
69
38. x2 - 4y2 = 1
36. y =
(g) f12x2
(h) f1x + h2
x2 - 1
42. f1x2 =
x + 4
46. f1x2 = 1 -
1
1x + 222
In Problems 47–60, find the domain of each function.
47. f1x2 = -5x + 4
51. g1x2 =
x
x2 - 16
48. f1x2 = x2 + 2
52. h1x2 =
49. f1x2 =
2x
53. F1x2 =
x2 - 4
55. h1x2 = 23x - 12
56. G1x2 = 21 - x
2
59. p1x2 =
Ax - 1
60. q1x2 = 2 -x - 2
57. f1x2 =
x
50. f1x2 =
2
x + 1
x - 2
54. G1x2 =
x3 + x
4
58. f1x2 =
2x - 9
x2
2
x + 1
x + 4
x3 - 4x
x
2x - 4
In Problems 61–70, for the given functions f and g, find the following functions and state the domain of each.
f
(a) f + g
(b) f - g
(c) f # g
(d)
g
61. f1x2 = 3x + 4; g1x2 = 2x - 3
62. f1x2 = 2x + 1; g1x2 = 3x - 2
63. f1x2 = x - 1; g1x2 = 2x2
64. f1x2 = 2x2 + 3; g1x2 = 4x3 + 1
65. f1x2 = 1x;
66. f1x2 = ƒ x ƒ ;
g1x2 = 3x - 5
1
1
67. f1x2 = 1 + ; g1x2 =
x
x
2x + 3
4x
69. f1x2 =
; g1x2 =
3x - 2
3x - 2
g1x2 = x
68. f1x2 = 2x - 2; g1x2 = 24 - x
70. f1x2 = 2x + 1; g1x2 =
71. Given f1x2 = 3x + 1 and 1f + g21x2 = 6 -
1
x, find the
2
72. Given
function g.
f1x2 =
1
x
and
2
x
f
x + 1
a b1x2 = 2
,
g
x - x
find
the
function g.
In Problems 73–78, find the difference quotient of f, that is, find
f1x + h2 - f1x2
73. f1x2 = 4x + 3
74. f1x2 = -3x + 1
76. f1x2 = x2 + 5x - 1
77. f1x2 = x3 - 2
h
, h Z 0, for each function. Be sure to simplify.
75. f1x2 = x2 - x + 4
1
78. f1x2 =
x + 3
Applications and Extensions
79. If f1x2 = 2x3 + Ax2 + 4x - 5 and f122 = 5, what is the
value of A?
80. If f1x2 = 3x2 - Bx + 4 and f1-12 = 12, what is the value
of B?
3x + 8
81. If f1x2 =
and f102 = 2, what is the value of A?
2x - A
1
2x - B
82. If f1x2 =
and f122 = , what is the value of B?
3x + 4
2
2x - A
83. If f1x2 =
and f142 = 0, what is the value of A?
x - 3
Where is f not defined?
x - B
, f122 = 0, and f112 is undefined, what
x - A
are the values of A and B?
84. If f1x2 =
85. Geometry Express the area A of a rectangle as a function
of the length x if the length of the rectangle is twice its
width.
86. Geometry Express the area A of an isosceles right triangle
as a function of the length x of one of the two equal sides.
87. Constructing Functions Express the gross salary G of a
person who earns $10 per hour as a function of the number
x of hours worked.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
70
CHAPTER 2
Functions and Their Graphs
88. Constructing Functions Tiffany, a commissioned salesperson, earns $100 base pay plus $10 per item sold. Express her
gross salary G as a function of the number x of items sold.
89. Effect of Gravity on Earth If a rock falls from a height of
20 meters on Earth, the height H (in meters) after x seconds
is approximately
92. Cross-sectional Area The cross-sectional area of a beam
cut from a log with radius 1 foot is given by the function
A1x2 = 4x31 - x2 , where x represents the length, in feet,
of half the base of the beam. See the figure. Determine the
cross-sectional area of the beam if the length of half the
base of the beam is as follows:
(a) One-third of a foot
(b) One-half of a foot
(c) Two-thirds of a foot
H1x2 = 20 - 4.9x2
(a) What is the height of the rock when x = 1 second?
x = 1.1 seconds? x = 1.2 seconds? x = 1.3 seconds?
(b) When is the height of the rock 15 meters? When is it 10
meters? When is it 5 meters?
(c) When does the rock strike the ground?
90. Effect of Gravity on Jupiter If a rock falls from a height of
20 meters on the planet Jupiter, its height H (in meters)
after x seconds is approximately
A (x ) 4x 1 x 2
1
x
H1x2 = 20 - 13x2
(a) What is the height of the rock when x = 1 second?
x = 1.1 seconds? x = 1.2 seconds?
(b) When is the height of the rock 15 meters? When is it 10
meters? When is it 5 meters?
(c) When does the rock strike the ground?
93. Economics The participation rate is the number of people
in the labor force divided by the civilian population (excludes military). Let L1x2 represent the size of the labor
force in year x and P1x2 represent the civilian population in
year x. Determine a function that represents the participation rate R as a function of x.
94. Crimes Suppose that V1x2 represents the number of violent crimes committed in year x and P1x2 represents the
number of property crimes committed in year x. Determine
a function T that represents the combined total of violent
crimes and property crimes in year x.
91. Cost of Trans-Atlantic Travel A Boeing 747 crosses the
Atlantic Ocean (3000 miles) with an airspeed of 500 miles
per hour. The cost C (in dollars) per passenger is given by
C1x2 = 100 +
36,000
x
+
10
x
where x is the ground speed 1airspeed ; wind2.
(a) What is the cost per passenger for quiescent (no wind)
conditions?
(b) What is the cost per passenger with a head wind of 50
miles per hour?
(c) What is the cost per passenger with a tail wind of 100
miles per hour?
(d) What is the cost per passenger with a head wind of 100
miles per hour?
95. Health Care Suppose that P(x) represents the percentage
of income spent on health care in year x and I1x2 represents
income in year x. Determine a function H that represents
total health care expenditures in year x.
96. Income Tax Suppose that I1x2 represents the income of
an individual in year x before taxes and T1x2 represents the
individual’s tax bill in year x. Determine a function N that
represents the individual’s net income (income after taxes)
in year x.
97. Some functions f have the property that f1a + b2 =
f1a2 + f1b2 for all real numbers a and b. Which of the following functions have this property?
(a) h1x2 = 2x
(b) g1x2 = x2
1
(c) F1x2 = 5x - 2
(d) G1x2 =
x
Discussion and Writing
x2 - 1
98. Are the functions f1x2 = x - 1 and g1x2 =
the
x + 1
same? Explain.
99. Investigate when, historically, the use of the function
notation y = f1x2 first appeared.
‘Are You Prepared?’ Answers
1. 1-1, 32
2. 21.5
3. 5x ƒ x Z -46
4. 5x ƒ x 6 -16
1
0
1
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall