Earlier we noted that the absolute value of a real number x can be defined as
Notice that this function is defined by different rules for different parts of its domain. Functions whose definitions involve more than one rule are called piecewise-defined functions. Graphing one of these functions involves graphing
x = e -x if x 6 0 x if x Ú 0
EXAMPLE 6 Natural Gas Rates Easton Utilities uses the rates shown in Table 1 to compute the monthly cost of natural gas for each customer. Write a piecewise definition for the cost of consuming x CCF (cubic hundred feet) of natural gas and graph the function.
each rule over the appropriate portion of the domain (Fig. 10). In Figure 10C, notice that an open dot is used to show that the point is not part of the graph and a solid dot is used to show that (0, 2) is part of the graph.
(0, -2)
x y
5
5 5
5
x y
5
5 5
5
x y
5
5 5
5
(A) y x2 2 (B) y 2 x2 (C) y x2 2 if x 0
2 x2 if x 0 Figure 10 Graphing a piecewise-defined function
Table 1 Charges per Month
$0.7866 per CCF for the first 5 CCF
$0.4601 per CCF for the next 35 CCF
$0.2508 per CCF for all over 40 CCF
SOLUTION If C(x) is the cost, in dollars, of using x CCF of natural gas in one month, then the first line of Table 1 implies that
Note that is the cost of 5 CCF. If then
repre-sents the amount of gas that cost $0.4601 per CCF, represents the cost of this gas, and the total cost is
If then
where the cost of the first 40 CCF. Combining all these equa-tions, we have the following piecewise definition for C(x):
To graph C, first note that each rule in the definition of C represents a transfor-mation of the identity function Graphing each transformation over the indicated interval produces the graph of C shown in Figure 11.
f(x) = x.
C(x) = *
0.7866x if 0 x 5
3.933 + 0.46501(x - 5) if 5 6 x 40 20.0365 + 0.2508(x - 40) if x 7 40 20.0365 = C(40),
C(x) = 20.0365 + 0.2508(x - 40) x 7 40,
C(x) = 3.933 + 0.4601(x - 5)
0.4601(x - 5)
x - 5 5 6 x 40,
C(5) = 3.933
C(x) = 0.7866x if 0 x 5
20
10 30 40 50 60 x
C(x)
$10
$20
$30
(5, 3.933)
(40, 20.0365)
Figure 11 Cost of purchasing x CCF of natural gas
As the next example illustrates, piecewise-defined functions occur naturally in many applications.
Matched Problem 6 Trussville Utilities uses the rates shown in Table 2 to compute the monthly cost of natural gas for residential customers. Write a piecewise definition for the cost of consuming x CCF of natural gas and graph the function.
Table 2 Charges per Month
$0.7675 per CCF for the first 50 CCF
$0.6400 per CCF for the next 150 CCF
$0.6130 per CCF for all over 200 CCF
Exercise 2-2
In Problems 1 8, give the domain and range of each function.
1. 2.
3. 4.
5. 6.
7. 8.
In Problems 9 20, graph each of the functions using the graphs of functions f and g below.
s(x) = 1 + 1x3
In Problems 21 28, indicate verbally how the graph of each func-tion is related to the graph of one of the six basic funcfunc-tions in Figure 1 on page 60. Sketch a graph of each function.
21. 22.
23. 24.
25. 26.
27. 28.
Each graph in Problems 29 36 is the result of applying a sequence of transformations to the graph of one of the six basic
m(x) = -0.4x2
functions in Figure 1 on page 60. Identify the basic function and describe the transformation verbally. Write an equation for the given graph.
In Problems 37 42, the graph of the function g is formed by ap-plying the indicated sequence of transformations to the given function f. Find an equation for the function g and graph g using
and .
37. The graph of is shifted 2 units to the right and 3 units down.
38. The graph of is shifted 3 units to the left and 2 units up.
39. The graph of is reflected in the x axis and shifted to the left 3 units.
40. The graph of is reflected in the x axis and shifted to the right 1 unit.
41. The graph of is reflected in the x axis and shifted 2 units to the right and down 1 unit.
42. The graph of is reflected in the x axis and shifted to the left 2 units and up 4 units.
Graph each function in Problems 43 48.
43.
Each of the graphs in Problems 49 54 involves a reflection in the x axis and/or a vertical stretch or shrink of one of the basic func-tions in Figure 1 on page 60. Identify the basic function, and describe the transformation verbally. Write an equation for the given graph.
Changing the order in a sequence of transformations may change the final result. Investigate each pair of transformations in Prob-lems 55 60 to determine if reversing their order can produce a dif-ferent result. Support your conclusions with specific examples and/or mathematical arguments.
55. Vertical shift; horizontal shift 56. Vertical shift; reflection in y axis 57. Vertical shift; reflection in x axis 58. Vertical shift; vertical stretch 59. Horizontal shift; reflection in y axis 60. Horizontal shift; vertical shrink
Applications
61. Price demand. A retail chain sells DVD players. The retail price p(x) (in dollars) and the weekly demand x for a partic-ular model are related by
(A) Describe how the graph of function p can be obtained from the graph of one of the basic functions in Figure 1 on page 60.
(B) Sketch a graph of function p using part (A) as an aid.
62. Price supply. The manufacturers of the DVD players in Problem 61 are willing to supply x players at a price of p(x) as given by the equation
(A) Describe how the graph of function p can be obtained from the graph of one of the basic functions in Figure 1 on page 60.
(B) Sketch a graph of function p using part (A) as an aid.
p(x) = 41x 9 x 289
Table 3 Summer (July October)
Base charge, $8.50
First 700 kWh or less at 0.0650/kWh Over 700 kWh at 0.0900/kWh
Table 4 Winter (November June)
Base charge, $8.50
First 700 kWh or less at 0.0650/kWh Over 700 kWh at 0.0530/kWh
Table 5 Kansas State Income Tax
SCHEDULE I MARRIED FILING JOINT If taxable income is
Over But Not Over Tax Due Is
$0 $30,000 3.50% of taxable income
$30,000 $60,000 $1,050 plus 6.25% of
excess over $30,000
$60,000 $2,925 plus 6.45% of
excess over $60,000
Table 6 Kansas State Income Tax
SCHEDULE II SINGLE, HEAD OF HOUSEHOLD, OR MARRIED FILING SEPARATE
If taxable income is
Over But Not Over Tax Due Is
$0 $15,000 3.50% of taxable income
$15,000 $30,000 $525 plus 6.25% of excess over $15,000
$30,000 $1,462.50 plus 6.45% of
excess over $30,000
66. Electricity rates. Table 4 shows the electricity rates charged by Monroe Utilities in the winter months.
(A) Write a piecewise definition of the monthly charge W(x) for a customer who uses x kWh in a winter month.
(B) Graph W(x).
67. State income tax. Table 5 shows a recent state income tax schedule for married couples filing a joint return in Kansas.
(A) Write a piecewise definition for the tax due T(x) on an income of x dollars.
(B) Graph T(x).
(C) Find the tax due on a taxable income of $40,000. Of
$70,000.
(A) Write a piecewise definition for the tax due T(x) on an income of x dollars.
(B) Graph T(x).
(C) Find the tax due on a taxable income of $20,000. Of
$35,000.
(D) Would it be better for a married couple in Kansas with two equal incomes to file jointly or separately? Discuss.
69. Human weight. A good approximation of the normal weight of a person 60 inches or taller but not taller than 80 inches is given by where x is height in inches and w(x)is weight in pounds.
w(x) = 5.5x - 220,
(A) Describe how the graph of function can be obtained from the graph of one of the basic functions in Figure 1, page 60.
(B) Sketch a graph of function using part (A) as an aid.
70. Herpetology. The average weight of a particular species of
snake is given by , , where x is
length in meters and (x) is weight in grams.w
0.2 x 0.8
w(x)= 463x3 w
w 63. Hospital costs. Using statistical methods, the financial
department of a hospital arrived at the cost equation
where C(x) is the cost in dollars for handling x cases per month.
(A) Describe how the graph of function C can be obtained from the graph of one of the basic functions in Figure 1 on page 60.
(B) Sketch a graph of function C using part (A) and a graphing calculator as aids.
64. Price demand. A company manufactures and sells in-line skates. Its financial department has established the price demand function
where p(x) is the price at which x thousand pairs of in-line skates can be sold.
(A) Describe how the graph of function p can be obtained from the graph of one of the basic functions in Figure 1 on page 60.
(B) Sketch a graph of function p using part (A) and a graphing calculator as aids.
65. Electricity rates. Table 3 shows the electricity rates charged by Monroe Utilities in the summer months. The base is a fixed monthly charge, independent of the kWh (kilowatt-hours) used during the month.
(A) Write a piecewise definition of the monthly charge
68. State income tax. Table 6 shows a recent state income tax schedule for individuals filing a return in Kansas.
(A) Describe how the graph of function can be obtained from the graph of one of the basic functions in Figure 1, page 60.
(B) Sketch a graph of function using part (A) as an aid.
71. Safety research. Under ideal conditions, if a person driving a vehicle slams on the brakes and skids to a stop, the speed of the vehicle (x) (in miles per hour) is given approximately by , where x is the length of skid marks (in feet) and C is a constant that depends on the road conditions and the weight of the vehicle. For a particular vehicle,
and .
(A) Describe how the graph of function can be obtained from the graph of one of the basic functions in Figure 1, page 60.
(B) Sketch a graph of function using part (A) as an aid.
72. Learning. A production analyst has found that on average it takes a new person T(x) minutes to perform a particular assembly operation after x performances of the operation,
where .
(A) Describe how the graph of function T can be obtained from the graph of one of the basic functions in Figure 1, page 60.
(B) Sketch a graph of function T using part (A) as an aid.
Answers to Matched Problems 1. (A)
(B)
2. (A) The graph of is the same as the graph of shifted upward 5 units, and the graph of
is the same as the graph of
shifted downward 4 units. The figure confirms these conclusions.
4. (A) The graph of is a vertical stretch of the graph of and the graph of is a vertical shrink of the graph of The figure confirms these conclusions. y = x.
y = 0.5x re-flection in the x axis of the graph of The figure confirms this conclusion.
y = x.
y = -0.5x
5. The graph of function G is a reflection in the x axis and a horizontal translation of 2 units to the left of the graph of
An equation for G is
6. C(x) = *