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Chapter Review Library of Functions Linear function (p. 109)
Constant function (p. 110)
f1x2 = mx + b
f1x2 = b
Graph is a line with slope m and y-intercept b.
Graph is a horizontal line with y-intercept b.
f (x ) mx b , m 0
y
y f (x ) = b
(0, b)
(0,b)
x
x
Identity function (p. 110)
Square function (p. 110)
Cube function (p. 110)
f1x2 = x
f1x2 = x2
f1x2 = x3
Graph is a line with slope 1 and y-intercept 0.
Graph is a parabola with intercept at 10, 02. y
y 3
( – 2, 4)
y 4
(2, 4)
4
(1, 1) (0, 0)
–3 (– 1, –1)
(1, 1) (– 1, 1)
3 x
(1, 1) 4 x
(0, 0)
–4
4 (1, 1)
(0, 0)
4
x
4
Square root function (p. 110)
Cube root function (p. 111)
Reciprocal function (p. 112)
f1x2 = 1x
f1x2 = 1 3x
f1x2 = y 3
y 2
1
1 x
(1, 1)
(0, 0)
y 2
(4, 2)
3
5 x
(1, 1)
( 1–8, 1–2)
(2, 2 )
( 1–8 , 1–2)
3
3 x (0, 0) 3
(1, 1) 2
(1, 1)
(2, 2 )
2 x
(1, 1)
3
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2
Chapter Review
Absolute value function (p. 111)
Greatest integer function (pp. 111–112)
f1x2 = ƒ x ƒ
f1x2 = int1x2 y
y 4
3 (2, 2)
(2, 2) (1, 1) 3
139
2
(1, 1)
(0, 0)
2
3 x
2
4
x
3
Things to Know Function (pp. 56–61)
A relation between two sets of real numbers so that each number x in the first set, the domain, has corresponding to it exactly one number y in the second set. The range is the set of y values of the function for the x values in the domain. x is the independent variable; y is the dependent variable.
A function can also be characterized as a set of ordered pairs 1x, y2 in which no first element is paired with two different second elements.
Function notation (pp. 61–64)
y = f1x2 f is a symbol for the function. x is the argument, or independent variable. y is the dependent variable. f1x2 is the value of the function at x, or the image of x. A function f may be defined implicitly by an equation involving x and y or explicitly by writing y = f1x2.
Difference quotient of f (p. 63 and p. 92)
f1x + h2 - f1x2 h
,
h Z 0
Domain (p. 58 and p. 64)
If unspecified, the domain of a function f is the largest set of real numbers for which f1x2 is a real number.
Vertical-line test (p. 72)
A set of points in the plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.
Even function f (p. 80)
f1-x2 = f1x2 for every x in the domain ( -x must also be in the domain).
Odd function f (p. 80)
f1-x2 = -f1x2 for every x in the domain ( -x must also be in the domain).
Increasing function (p. 83)
A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 6 x2 , we have f1x12 6 f1x22.
Decreasing function (p. 83)
A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 6 x2 , we have f1x12 7 f1x22.
Constant function (p. 83)
A function f is constant on an interval I if, for all choices of x in I, the values of f1x2 are equal.
Local maximum (p. 84)
A function f has a local maximum at c if there is an open interval I containing c so that, for all x in I, f1x2 … f1c2.
Local minimum (p. 84)
A function f has a local minimum at c if there is an open interval I containing c so that, for all x in I, f1x2 Ú f1c2.
Average rate of change of a function (p. 85)
The average rate of change of f from c to x is f1x2 - f1c2 ¢y = , ¢x x - c
Linear function (p. 93)
x Z c
f1x2 = mx + b Graph is a line with slope m and y-intercept b.
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Objectives Section 2.1
2.2
2.3
You should be able to
1 ✓ 2 ✓ 3 ✓ 4 ✓ 1 ✓ 2 ✓ 1 ✓ 2 ✓ 3 ✓ 4 ✓ 5 ✓
2.6
6 ✓ 1 ✓ 2 ✓ 3 ✓ 4 ✓ 5 ✓ 6 ✓ 1 ✓ 2 ✓ 1 ✓
2.7
2 ✓ 3 ✓ 1 ✓
2.4
2.5
Review Exercises
Á
Determine whether a relation represents a function (p. 56)
1, 2, 81(a)
Find the value of a function (p. 61)
3–8, 23, 24, 73, 74
Find the domain of a function (p. 64)
9–16, 81(f)
Form the sum, difference, product, and quotient of two functions (p. 65)
17–22
Identify the graph of a function (p. 71)
47, 48
Obtain information from or about the graph of a function (p. 72)
25(a)–(e), 26(a)–(e), 27(a), 27(f), 28(a), 28(f)
Determine even and odd functions from a graph (p. 80)
27(e), 28(e)
Identify even and odd functions from the equation (p. 81)
29–36
Use a graph to determine where a function is increasing, decreasing, or constant (p. 82)
27(b), 28(b)
Use a graph to locate local maxima and local minima (p. 83)
27(c), 28(c)
Use a graphing utility to approximate local maxima and local minima and to determine where a function is increasing or decreasing (p. 84)
37–40, 87
Find the average rate of change of a function (p. 85)
41–46
Graph linear functions (p. 93)
49–52
Work with applications of linear functions (p. 94)
75, 76, 79
Draw and interpret scatter diagrams (p. 96)
81(b)
Distinguish between linear and nonlinear relations (p. 98)
81(b)
Use a graphing utility to find the line of best fit (p. 98)
81(c)
Construct a linear model using direct variation (p. 100)
82, 83
Graph the functions listed in the library of functions (p. 109)
53, 54
Graph piecewise-defined functions (p. 112)
67–70
Graph functions using vertical and horizontal shifts (p. 118)
25(f), 26(f), 26(g) 55, 56, 59–66
Graph functions using compressions and stretches (p. 120)
25(g), 26(h), 57, 58, 65, 66
Graph functions using reflections about the x-axis or y-axis (p. 123)
25(h), 57, 61, 62, 66
Construct and analyze functions (p. 130)
78, 80, 84–86, 88–92
Review Exercises In Problems 1 and 2, determine whether each relation represents a function. For each function, state the domain and range. 1. 51-1, 02, 12, 32, 14, 026 2. 514, -12, 12, 12, 14, 226 In Problems 3–8, find the following for each function: (a) f122 3. f1x2 =
(b) f1-22 3x
x2 - 1
6. f1x2 = ƒ x2 - 4 ƒ
(c) f1-x2
(d) -f1x2
(e) f1x - 22
x2 4. f1x2 = x + 1 x2 - 4 7. f1x2 = x2
In Problems 9–16, find the domain of each function. x 3x2 9. f1x2 = 2 10. f1x2 = x - 2 x - 9 ƒxƒ 1x 13. h1x2 = 14. g1x2 = x ƒxƒ
(f) f12x2
5. f1x2 = 3x2 - 4 8. f1x2 =
11. f1x2 = 22 - x 15. f1x2 =
x x2 + 2x - 3
x3 x2 - 9
12. f1x2 = 2x + 2 16. F1x2 =
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1 x2 - 3x - 4
Chapter Review
In Problems 17–22, find f + g, f - g, f # g, and
f for each pair of functions. State the domain of each. g
17. f1x2 = 2 - x; g1x2 = 3x + 1
18. f1x2 = 2x - 1; g1x2 = 2x + 1
20. f1x2 = 3x; g1x2 = 1 + x + x2
21. f1x2 =
19. f1x2 = 3x2 + x + 1; g1x2 = 3x
x + 1 1 ; g1x2 = x - 1 x
22. f1x2 =
1 3 ; g1x2 = x - 3 x
In Problems 23 and 24, find the difference quotient of each function f; that is, find f1x + h2 - f1x2 h
,
h Z 0
23. f1x2 = -2x2 + x + 1
24. f1x2 = 3x2 - 2x + 4
25. Using the graph of the function f shown: (a) Find the domain and the range of f.
26. Using the graph of the function g shown: (a) Find the domain and the range of g.
(b) List the intercepts.
(b) Find g1-12.
(c) Find f1-22.
(c) List the intercepts.
(d) For what values of x does f1x2 = -3?
(d) For what value of x does g1x2 = -3?
(e) Solve f1x2 7 0.
(e) Solve g1x2 7 0.
(f) Graph y = f1x - 32.
(f) Graph y = g1x - 22.
1 (g) Graph y = fa xb . 2
(g) Graph y = g1x2 + 1. (h) Graph y = 2g1x2.
(h) Graph y = -f1x2. y 4
(5, 1)
(3, 3)
(4, 0) 5
5 (2, 1)
(0, 0)
(4, 3)
y 3 (1, 1)
5
3
x
4
In Problems 27 and 28, use the graph of the function f to find: (a) The domain and the range of f (b) The intervals on which f is increasing, decreasing, or constant (c) The local minima and local maxima (d) Whether the graph is symmetric with respect to the x-axis, the y-axis, or the origin (e) Whether the function is even, odd, or neither (f) The intercepts, if any 27.
28.
y
(2, 1)
3 (1, 1)
(0, 0)
(4, 0)
5 (4, 2)
5 x 3
y 4
6 (4,3) (3, 0)
(3, 0)
5
(0, 0)
(4, 3)
6 x (2, 1)
4
(3, 3)
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(3, 3)
x
141
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Functions and Their Graphs
In Problems 29–36, determine (algebraically) whether the given function is even, odd, or neither. 4 + x2
29. f1x2 = x3 - 4x
30. g1x2 =
33. G1x2 = 1 - x + x3
34. H1x2 = 1 + x + x2
31. h1x2 =
1 + x4
35. f1x2 =
1
1 +
x4
x2
32. F1x2 = 31 - x3
+ 1
x 1 + x
36. g1x2 =
2
1 + x2 x3
In Problems 37–40, use a graphing utility to graph each function over the indicated interval. Approximate any local maxima and local minima. Determine where the function is increasing and where it is decreasing. 37. f1x2 = 2x3 - 5x + 1 1-3, 32
38. f1x2 = -x3 + 3x - 5 1-3, 32
39. f1x2 = 2x4 - 5x3 + 2x + 1 1-2, 32
40. f1x2 = -x4 + 3x3 - 4x + 3 1-2, 32
In Problems 41 and 42, find the average rate of change of f: (a) From 1 to 2 (b) From 0 to 1 (c) From 2 to 4 41. f1x2 = 8x2 - x
42. f1x2 = 2x3 + x
In Problems 43–46, find the average rate of change from 2 to x for each function f. Be sure to simplify. 44. f1x2 = 2x2 + 7
43. f1x2 = 2 - 5x
45. f1x2 = 3x - 4x2
46. f1x2 = x2 - 3x + 2
In Problems 47 and 48, tell which of the following graphs are graphs of functions. 47.
y
48.
y
x
y
y
x
x
(a)
(b)
(a)
x
(b)
In Problems 49–52, graph each linear function. 49. f1x2 = 2x - 5
50. g1x2 = -4x + 7
51. h1x2 =
4 x - 6 5
1 52. F1x2 = - x + 1 3
In Problems 53 and 54, sketch the graph of each function. Be sure to label at least three points. 54. f1x2 = 1 3x
53. f1x2 = ƒ x ƒ
In Problems 55–66, graph each function using the techniques of shifting, compressing or stretching, and reflections. Identify any intercepts on the graph. State the domain and, based on the graph, find the range. 1 ƒxƒ 2
55. F1x2 = ƒ x ƒ - 4
56. f1x2 = ƒ x ƒ + 4
57. g1x2 = -2 ƒ x ƒ
58. g1x2 =
59. h1x2 = 2x - 1
60. h1x2 = 1x - 1
61. f1x2 = 21 - x
62. f1x2 = - 2x + 3
63. h1x2 = 1x - 122 + 2
64. h1x2 = 1x + 222 - 3
65. g1x2 = 31x - 123 + 1
66. g1x2 = -21x + 223 - 8
In Problems 67–70, (a) Find the domain of each function. (c) Graph each function. 67. f1x2 = b
3x x + 1
x 69. f1x2 = c 1 3x
-2 6 x … 1 x 7 1 -4 … x 6 0 x = 0 x 7 0
(b) Locate any intercepts. (d) Based on the graph, find the range. 68. f1x2 = b
x - 1 3x - 1
-3 6 x 6 0 x Ú 0
70. f1x2 = b
x2 2x - 1
-2 … x … 2 x 7 2
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Chapter Review
71. Given that f is a linear function, f142 = -5, and f102 = 3, write the equation that defines f. 72. Given that g is a linear function with slope = -4 and g1-22 = 2, write the equation that defines g. 73. A function f is defined by f1x2 =
Ax + 5 6x - 2
143
(e) Use TRACE to determine what margin should be used to obtain an area of 70 square inches and of 50 square inches. 79. Strength of a Beam The strength of a rectangular wooden beam is proportional to the product of the width and the cube of its depth (see the figure). If the beam is to be cut from a log in the shape of a cylinder of radius 3 feet, express the strength S of the beam as a function of the width x. What is the domain of S?
If f112 = 4, find A.
g1x2 =
A 8 + 2 x x
If g1 -12 = 0, find A. 75. Temperature Conversion The temperature T of the air is approximately a linear function of the altitude h for altitudes within 10,000 meters of the surface of Earth. If the surface temperature is 30°C and the temperature at 10,000 meters is 5°C, find the function T = T1h2. 76. Speed as a Function of Time The speed v (in feet per second) of a car is a linear function of the time t (in seconds) for 10 … t … 30. If after each second the speed of the car has increased by 5 feet per second and if after 20 seconds the speed is 80 feet per second, how fast is the car going after 30 seconds? Find the function v = v1t2. 77. Spheres The volume V of a sphere of radius r is 4 V = pr3; the surface area S of this sphere is S = 4pr2. 3 If the radius doubles, how does the volume change? How does the surface area change? 1 A page with dimensions of 8 inches by 11 2 inches has a border of uniform width x surrounding the printed matter of the page, as shown in the figure.
78. Page Design
1
8 –2 in. x The most important Beatle album to come out in 1968 was simply entitled The Beatles. It has become known as the “White Album” because its cover is completely white and devoid of any front or graphics except on the spine and a number on the front cover representing the order of production. Having launched an explosion of garish, elaborate album art with Sgt. Pepper, the Beatles now went to the opposite extreme with the ultimate in plain simplicity. The White Album was a double album (previously rare
x
x
11 in.
in pop music except for special collections) and contained thirty songs. Beatle fans consider it either their heroes’ best or worst album! The controversy arises from the extreme eclecticism of the music: there is a bewildering variety of styles on this album. Although the reason for for this eclecticism was not apparent at the time, it has since become obvious. The White Album was not so much the work of one group but four individuals each of whom was heading in a different direction.
x
(a) Write a formula for the area A of the printed part of the page as a function of the width x of the border. (b) Give the domain and the range of A. (c) Find the area of the printed the page for borders of widths 1 inch, 1.2 inches, and 1.5 inches. (d) Graph the function A = A1x2.
Width, x
74. A function g is defined by
3 ft
Depth
80. Material Needed to Make a Drum A steel drum in the shape of a right circular cylinder is required to have a volume of 100 cubic feet. (a) Express the amount A of material required to make the drum as a function of the radius r of the cylinder. (b) How much material is required if the drum is of radius 3 feet? (c) How much material is required if the drum is of radius 4 feet? (d) How much material is required if the drum is of radius 5 feet? (e) Graph A = A1r2. For what value of r is A smallest? 81. High School versus College GPA An administrator at Southern Illinois University wants to find a function that relates a student’s college grade point average G to the high school grade point average x. She randomly selects eight students and obtains the following data: High School GPA, x
College GPA, G
2.73
2.43
2.92
2.97
3.45
3.63
3.78
3.81
2.56
2.83
2.98
2.81
3.67
3.45
3.10
2.93
(a) Does the relation defined by the set of ordered pairs 1x, G2 represent a function? (b) Draw a scatter diagram of the data. Are the data linear? (c) Using a graphing utility, find the line of best fit relating high school GPA and college GPA.
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CHAPTER 2
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(d) Interpret the slope. (e) Express the relationship found in part (c) using function notation. (f) What is the domain of the function? (g) Predict a student’s college GPA if her high school GPA is 3.23. 82. Mortgage Payments The monthly payment p on a mortgage varies directly with the amount borrowed B. If the monthly payment on a 30-year mortgage is $854.00 when $130,000 is borrowed, find a linear function that relates the monthly payment p to the amount borrowed B for a mortgage with the same terms. Then find the monthly payment p when the amount borrowed B is $165,000. 83. Revenue Function At the corner Esso station, the revenue R varies directly with the number g of gallons of gasoline sold. If the revenue is $28.89 when the number of gallons sold is 13.5, find a linear function that relates revenue R to the number g of gallons of gasoline. Then find the revenue R when the number of gallons of gasoline sold is 11.2. 84. Landscaping A landscape engineer has 200 feet of border to enclose a rectangular pond. What dimensions will result in the largest pond? 85. Geometry Find the length and width of a rectangle whose perimeter is 20 feet and whose area is 16 square feet. 86. A rectangle has one vertex on the line y = 10 - x, x 7 0, another at the origin, one on the positive x-axis, and one on the positive y-axis. Find the largest area A that can be enclosed by the rectangle. 87. Minimizing Marginal Cost The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the 50th product is $6.20, then it cost $6.20 to increase production from 49 to 50 units of output. Callaway Golf Company has determined that the marginal cost C of manufacturing x Big Bertha golf clubs may be expressed by the quadratic function C1x2 = 4.9x2 - 617.4x + 19,600 (a) How many clubs should be manufactured to minimize the marginal cost? (b) At this level of production, what is the marginal cost?
88. Find the point on the line y = x that is closest to the point 13, 12. 89. Find the point on the line y = x + 1 that is closest to the point 14, 12. 90. A rectangle has one vertex on the graph of y = 10 - x2, x 7 0, another at the origin, one on the positive x-axis, and one on the positive y-axis. Find the largest area A that can be enclosed by the rectangle. 91. Constructing a Closed Box A closed box with a square base is required to have a volume of 10 cubic feet. (a) Express the amount A of material used to make such a box as a function of the length x of a side of the square base. (b) How much material is required for a base 1 foot by 1 foot? (c) How much material is required for a base 2 feet by 2 feet? (d) Graph A = A1x2. For what value of x is A smallest? 92. Cost of a Drum A drum in the shape of a right circular cylinder is required to have a volume of 500 cubic centimeters. The top and bottom are made of material that costs 6¢ per square centimeter; the sides are made of material that costs 4¢ per square centimeter.
500 cc
(a) Express the total cost C of the material as a function of the radius r of the cylinder. (b) What is the cost if the radius is 4 cm? (c) What is the cost if the radius is 8 cm? (d) Graph C = C1r2. For what value of r is the cost C least?
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