5 Function Notation: The Concept of Function as a Rule
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SECTION 1.5
■
Function Notation: The Concept of Function as a Rule
53
Function Notation
A function f is a rule that assigns to each input exactly one output. If we
write x for the input and y for the output, then we use the following notation
to describe f :
Function Input Output
T
T T
f 1x2 = y
The symbol f 1x2 is read “f of x” or “f at x” and is called the value of f at x or the image of x under f. This notation emphasizes the dependence of the output on the corresponding input—namely, the output f 1x2 is the result of applying the rule f to the
input x:
x S f 1x2
The following examples illustrate the meaning of function notation.
e x a m p l e 1 Function Notation
Consider the function f 1x2 = 8x.
(a) What is the name of the function?
(b) What letter represents the input? What is the output?
(c) What rule does this function represent?
(d) Find f 122 . What does f 122 represent?
Solution
(a)
(b)
(c)
(d)
The name of the function is f.
The input is x, and the output is 8x.
The rule is “Multiply the input by 8.”
f 12 2 = 8 ؒ 2 = 16. So 16 is the value of the function at 2.
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NOW TRY EXERCISE 9
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e x a m p l e 2 Writing Function Notation
Express the given rule in function notation.
(a) “Multiply by 2, then add 5.”
(b) “Add 3, then square.”
Solution
(a) First we need to choose a letter to represent this rule. So let g stand for the rule
“Multiply by 2, then add 5.” Then for any input x, multiplying by 2 gives 2x,
then adding 5 gives 2x + 5. Thus we can write
g1x2 = 2x + 5
Note that the input is the number x and the corresponding output is the number
2x + 5.
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Data, Functions, and Models
(b) Let’s choose the letter h to stand for the rule “Add 3, then square.” Then for
any input x, adding 3 gives x + 3, then squaring gives 1x + 32 2. Thus we
have
h 1x2 = 1x + 32 2
Note that the input is the number x and the corresponding output is the number
1x + 32 2.
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NOW TRY EXERCISE 13
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e x a m p l e 3 Expressing a Function in Words
Give a verbal description of the given function.
(a) f 1x2 = x - 5
(b) g1x2 =
x +3
2
Solution
(a) The rule f is “Subtract 5 from the input.”
(b) The rule g is “Add 3 to the input, then divide by 2.”
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NOW TRY EXERCISE 17
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e x a m p l e 4 Dependent and Independent Variables
(a) Express the equation y = x 2 + 2x in function notation, where x is the independent variable and y is the dependent variable.
(b) Express the function f 1x2 = 5x + 1 as an equation in two variables. Identify
the dependent and independent variables.
Solution
Although we chose the letter x for
the independent variable, any letter
can be chosen. The function (or
rule) is the same regardless of the
letter we choose to describe it.
(a) The equation determines a function in which each input x gives the output
x 2 + 2x. If we call this rule g, then we can write this function as
g1x2 = x 2 + 2x
(b) If we write y for the output f 1x 2 , then we can write the function as
y = 5x + 1
The dependent variable is y, and the independent variable is x.
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2
NOW TRY EXERCISES 23 AND 29
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■ Evaluating Functions—Net Change
In function notation the input x plays the role of a placeholder. For example, the function f 1x2 = 3x 2 + x can be thought of as
f 1 ٗ 2 = 31 ٗ 2 2 +
ٗ
SECTION 1.5
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Function Notation: The Concept of Function as a Rule
55
So any letter can be used for the input of a function. Both of the following are the
same function:
f 1x 2 = 3x 2 + x
f 1z 2 = 3z 2 + z
We can see that each of these represent the same rule applied to the input.
e x a m p l e 5 Evaluating a Function
Let f 1x2 = 3x 2 + x. Evaluate the function at the given input.
(a) f 1- 22
(b) f 10 2
(c) f 14 2
(d) f A 12 B
Solution
To evaluate f at a number, we substitute the number for x in the definition of f.
(a)
(b)
(c)
(d)
■
f 1- 22 = 3 # 1- 2 2 2 + 1- 22 = 10
f 10 2 = 3 # 102 2 + 0 = 0
f 14 2 = 3 # 142 2 + 4 = 52
f A 12 B = 3 # A 12 B 2 + 12 = 54
NOW TRY EXERCISE 39
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Function notation is a useful and practical way of describing the rule of a function; we’ll see many examples of the advantages of function notation in the coming
chapters. Here we’ll see how function notation allows us to write a concise formula
for the net change in the value of a function between two points.
Net Change in the Value of a Function
The net change in the value of a function f as x changes from a to b (where
a … b) is given by
f 1b2 - f 1a2
In the next example we find the net change in the weight of an astronaut at different elevations above the earth.
e x a m p l e 6 Net Change in the Weight of an Astronaut
If an astronaut weighs 130 pounds on the surface of the earth, then her weight when
she is h miles above the earth is given by the function
w 1h2 = 130 a
2
3960
b
3960 + h
(a) Evaluate w(100). What does your answer mean?
(b) Construct a table of values for the function w that gives the astronaut’s weight
at heights from 0 to 500 miles. What do you conclude from the table?
(c) Find the net change in the weight of the astronaut from an elevation of
100 miles to an elevation of 400 miles. Interpret your result.
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Data, Functions, and Models
Solution
(a) We want the value of the function w when h is 100; substituting 100 for h in
the definition of w, we get
w 1h2 = 130 a
w 11002 = 130 a
2
3960
b
3960 + h
Function
2
3960
b
3960 + 100
Replace h by 100
L 123.67
h
w(h)
0
100
200
300
400
500
130
124
118
112
107
102
Calculator
This means that at a height of 100 mi, she weighs about 124 lb.
(b) The table gives the astronaut’s weight, rounded to the nearest pound, at
100-mile increments. The values in the table in the margin are calculated as in
part (a). The table indicates that the higher the astronaut ascends, the less she
weighs.
(c) The net change in the weight of the astronaut between these two elevations is
w14002 - w11002 . We use the entries already calculated in the table in part (b)
to evaluate this net change.
w14002 - w11002 L 107 - 124
From the table
= - 17
The net change is - 17 lb. The negative sign means that the astronaut’s weight
decreased by about 17 lb.
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2
NOW TRY EXERCISES 35 AND 61
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■ The Domain of a Function
The domain of a function is the set of all inputs for the function. The domain may
be stated explicitly. For example, if we write
f 1x2 = x 2
0…x…5
then the domain is the set of real numbers x for which 0 … x … 5. If the domain of
a function is not given explicitly, then by convention the domain is the set of all real
number inputs for which the output is defined. For example, consider the functions
f 1x2 =
1
x -4
g1x 2 = 1x
The function f is not defined when x is 4, so its domain is 5x Έ x
is not defined for negative x, so its domain is 5x Έ x Ú 06.
e x a m p l e 7 Finding the Domain of a Function
Find the domain of each function.
(a) f 1x2 =
1
x1x - 22
(b) g1x2 = 1x - 1
46. The function g
SECTION 1.5
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Function Notation: The Concept of Function as a Rule
57
Solution
Solving inequalities is reviewed
in Algebra Toolkit C.3, page T62.
(a) The function is not defined when the denominator is 0, that is, when x is 0 or
2. So the domain of f is 5x Έ x 0 and x 26.
(b) The function is defined only when x - 1 Ú 0. This means that x Ú 1. So the
domain of g is 5x Έ x Ú 16.
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2
NOW TRY EXERCISES 49 AND 51
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■ Piecewise Defined Functions
A piecewise defined function is a function that is defined by different rules on different parts of its domain. The rules for such functions cannot be expressed as a
single algebraic equation. This is one of the many reasons that we need function
notation.
example 8
Cell Phone Plan
A cell phone plan has a basic charge of $39 per month. The plan includes 400 minutes and charges 20 cents for each additional minute of usage. The monthly charges
are a function of the number x of minutes used, given by
C 1x2 = e
39
39 + 0.201x - 4002
if 0 … x … 400
if x 7 400
Find (a) C 11002 , (b) C 14002 , and (c) C 14802 .
Solution
Remember that a function is a rule. Here is how we apply the rule for this function.
First we look at the value of the input x. If 0 … x … 400, then the value of C(x) is 39.
On the other hand, if x 7 400, the value of C(x) is 39 + 0.201x - 4002 .
(a) Since 100 … 400, we have C11002 = 39.
(b) Since 400 … 400, we have C14002 = 39.
(c) Since 480 7 400, we have C14802 = 39 + 0.201480 - 4002 = 55. Thus, the
plan charges $39 for 100 minutes, $39 for 400 minutes, and $55 for 480
minutes.
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Aaron Kohr/Shutterstock.com 2009
IN CONTEXT ➤
The California aqueduct
NOW TRY EXERCISES 55 AND 67
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Some of the fastest-growing cities in the United States are located in areas where
water is in short supply. In the Southwest, water is brought to many cities from distant rivers and lakes via aqueducts. City planners must regulate growth carefully to
ensure that adequate water sources exist for new developments. On average, each
U.S. resident uses between 40 and 80 gallons of water daily. In arid areas people tend
to use less water. In Arizona, for example, people conserve water in many ways, including the use of xeriscaping (landscaping with desert-friendly plants).
To discourage excessive water use in arid areas, cities charge different rates that
depend on the amount of water used. In the next example we model the domestic cost
of water using a piecewise defined function.
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Data, Functions, and Models
e x a m p l e 9 Water Rates
To discourage excessive water use, a city charges its residents $0.008 per gallon for
households that use less than 4000 gallons a month and $0.012 for households that
use 4000 gallons or more a month.
(a) Find a piecewise defined function C that gives the water bill for a household
using x gallons per month.
(b) Find C (3900) and C(4200). What do your answers represent?
Solution
(a) Since the cost of x gallons of water depends on the usage, we need to define
the function C in two pieces: for x 6 4000 and for x Ú 4000. For x gallons
the cost is 0.008x if x 6 4000 and 0.012x if x Ú 4000. So we can express the
function C as
C1x 2 = e
0.008x
0.012x
if x 6 4000
if x Ú 4000
(b) Since 3900 6 4000, we have C139002 = 0.008139002 = 31.20. Since
4200 7 4000, we have C142002 = 0.012142002 = 50.40. So using 3900
gallons costs $31.20; using 4200 gallons costs $50.40.
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NOW TRY EXERCISE 69
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1.5 Exercises
CONCEPTS
Fundamentals
1. If a function f is given by the equation y = f 1x 2 , then the independent variable is
_______, the dependent variable is _______, and f 1a2 is the _______ of f at a.
2. If f 1x2 = x 2 + 1, then f 122 = _______ and f 102 = _______.
3. The net change in the value of the function f from a to b is the difference ____ Ϫ ____.
So if f 1x2 = x 2 + 1, then the net change in the value of the function f when x changes
from 0 to 2 is the difference ____ Ϫ ____ ϭ ____.
4. For a function f, the set of all possible inputs is called the _______ of f, and the set of
all possible outputs is called the _______ of f.
5. (a) Which of the following functions have 5 in their domain?
f 1x 2 = x 2 - 3x
g 1x 2 =
x -5
x
h 1x 2 = 1x - 10
(b) For the functions from part (a) that do have 5 in their domain, find the value of the
function at 5.
x
f (x)
0
19
2
4
6
6. A function is given algebraically by the formula f 1x 2 = 1x - 4 2 2 + 3. Fill in the table
in the margin to give a numerical representation of f.
Think About It
7. How would you describe the quantity f 1b 2 - f 1a 2 without using function notation? For
example, how would you describe the net change between - 2 and 5 for the function
y = x 2 - 2 without function notation?
SECTION 1.5
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Function Notation: The Concept of Function as a Rule
59
8. True or false?
(a) If the net change of a function f from a to b is zero, then the function must be
constant between a and b.
(b) If the net change of a function f from a to b is positive, then the values of the
function must steadily increase from a to b.
SKILLS
9–12 ■
(a)
(b)
(c)
(d)
A function is given.
What is the name of the function?
What letter represents the input? What is the output?
What rule does this function represent?
Find f 1102 . What does f 1102 represent?
9. f 1x2 = 2x + 1
13–16
■
10. g1w 2 = w 3 - 1
11. h1z 2 = 1z - 2 2 2
12. A1r2 =
r2
-7
5
Express the given rule in function notation.
13. “Add 2, then multiply by 5.”
14. “Divide by 3, then add 2.”
15. “Square, add 7, then divide by 4.”
16. “Subtract 5, square, then subtract 1.”
17–22
■
Give a verbal description of the function.
17. f 1x2 =
18. g1x 2 =
x
+7
2
x+7
2
19. h1w2 = 7w 2 - 5
20. k1w2 = 1w + 12 2 - 9
21. A1r 2 = 51r + 32 3
22. V 1r 2 =
2
+1
r
23–28 ■ An equation is given.
(a) Does the equation define a function with x as the independent variable and y as the
dependent variable? If so, express the equation in function notation with x as the
independent variable.
(b) Does the equation define a function with y as the independent variable and x as the
dependent variable? If so, express the equation in function notation with y as the
independent variable.
23. y = 5x
24. y = 3x 3
25. x = 4y 3
26. x = y 3 - 1
27. y = 3x 2
28. x = 4y 2
29–34
■
A function is given. Express the function as an equation. Identify the independent
and dependent variables.
29. f 1x2 = 3x 2 - 1
31. f 1w2 = 3w 2 - 1
30. g1x 2 = 2x 3 - 2x
32. g1z 2 = 2z 3 - 2z
33. S1r 2 = 4pr 2
34. V 1r2 = 43pr 3
35–38 ■ A function is given.
(a) Complete the table of values for the function.
(b) Find the net change in the value of the function when x changes from 0 to 2.
35. f 1x2 = 2x 2 - 7
x
f(x)
-2
-1
36. g1s 2 = 3s - 5
0
1
2
s
g(s)
-4
-2
0
1
2
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Data, Functions, and Models
37. h1z 2 = 1z + 32 3 - 2
z
-3
-1
38. k1t2 =
0
1
t
2
h(z)
39–42
■
1
t +1
0
1
2
3
4
k(t)
Evaluate the function at the indicated values.
39. f 1x2 = x 2 + 2x:
3
(a) f 10 2
2
40. g1x2 = x - 4x :
(a) g12 2
(b) f 1 12 2
(c) f 1- 22
(d) f 1a 2
(b)
(c) g1- 12
(d) g1b2
gA 12 B
41. r1x2 = 12x - 1:
(a) r12 2
(b) r15 2
(c) r13 2
(d) r1c 2
42. s1x 2 =
(a) s10 2
(b) s13 2
(c) s14 2
(d) s1a 2
1
:
1x + 1
43. The surface area S of a sphere is a function of its radius r given by S1r2 = 4pr 2.
(a) Find S122 and S132 .
(b) What do your answers in part (a) represent?
(c) Find the net change in the value of the function S when r changes from 1 to 4.
44. The volume V of a can that has height two times the radius is a function of its radius r
given by V 1r 2 = 2pr 3.
(a) Find V 142 and V 192 .
(b) What do your answers in part (a) represent?
(c) Find the net change in the value of the function V when r changes from 1 to 4.
45–54
■
Find the domain of the function.
45. f 1x2 = 2x
46. f 1x2 = x 2 + 1
47. g1x2 = 2x, - 1 … x … 5
49. h1x2 =
1
x-3
48. g1x2 = x 2 + 1, 0 … x … 5
50. h1x2 =
1
3x - 6
51. k1x 2 = 1x - 5
52. k1x 2 = 1x + 9
53. r1x 2 =
54. r1x 2 =
55–60
■
3
1x - 4
12 + x
3 -x
Evaluate the piecewise defined function at the indicated values.
55. f 1x2 = e
if x 6 0
if x Ú 0
(b) f 10 2
(c) f 1- 22
(d) f 11 2
56. g1x2 = e
if x 6 0
if x Ú 0
(b) g10 2
(c) g1- 2 2
(d) g11 2
(c) h13 2
(d) h11 2
(c) k1- 32
(d) k17 2
-x
x
(a) f 1- 1 2
- 2x
2x
(a) g1- 12
57. h1x2 = e
2
x
x+1
(a) h1- 3 2
58. k1x 2 = e
(a) k122
5
2x - 3
if x 6 1
if x Ú 1
(b) h10 2
if x … 2
if x 7 2
(b) k10 2
SECTION 1.5
59. F 1x2 = e
(a) F 12 2
x2 + 2x
2x
■
Function Notation: The Concept of Function as a Rule
if x … - 1
if x 7 - 1
(b) F 1- 12
3x
if x 6 0
60. G 1x2 = c x + 1
if 0 … x … 2
2
if x 7 2
1x - 22
(a) G 1- 2 2
(b) G 1- 12
CONTEXTS
(c) F 10 2
(d) F 1- 22
(c) G 10 2
(d) G 13 2
61
61. How Far Can You See? Because of the curvature of the earth, the maximum distance
D that one can see from the top of a tall building or from an airplane at height h is
modeled by the function
D 1h2 = 27920h + h 2
where D and h are measured in miles.
(a) Find D(0.1). What does this value represent?
(b) How far can one see from the observation deck of Toronto’s CN Tower, 1135 ft
above the ground? (Remember that one mile is 5280 ft.)
(c) Commercial aircraft fly at an altitude of about 7 mile. How far can the pilot see?
(d) Find the net change of the distance D as one climbs the CN Tower from a height of
100 ft to a height of 1135 ft.
62. Torricelli’s Law A tank holds 50 gallons of water, which drains from a leak at the
bottom, causing the tank to empty in 20 minutes. The tank drains faster when it is
nearly full because the pressure on the leak is greater. Torricelli’s Law models the
volume V (in gallons) of water remaining in the tank after t minutes as
V 1t 2 = 50 a 1 -
t 2
b
20
0 … t … 20
(a) Find V102 and V1202 . What do these values represent?
(b) Make a table of values of V1t2 for t ϭ 0, 5, 10, 15, 20.
(c) Find the net change of the volume of water in the tank when t changes from 0 to 10.
63. A Falling Sky Diver When a sky diver jumps out of an airplane from a height of
13,000 ft, her height h (in feet) above the ground after t seconds is given by the function
h 1t2 = 13,000 - 16t 2
h(t)=13,000-16t™
(a) Find h 1102 and h 1202 . What do these values represent?
(b) For safety reasons a sky diver must open the parachute at a height of about 2500 ft
(or higher). A sky diver opens her parachute after 24 seconds. Did she open her
parachute at a safe height?
(c) Find the net change in the sky diver’s height from 0 to 25 seconds.
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Data, Functions, and Models
64. Path of a Ball
function
A baseball is thrown across a playing field. Its path is given by the
h 1x2 = - 0.005x 2 + x + 5
where x is the distance the ball has traveled horizontally and h 1x 2 is its height above
ground level, both measured in feet.
(a) Find h 1102 and h 11002 . What do these values represent?
(b) What is the initial height of the ball?
(c) Find the net change in the height of the ball as the horizontal distance x changes
from 10 to 100 ft.
x
(year)
R(x)
(billions of dollars)
2000
2001
2002
2003
2004
2005
2006
7.66
8.41
9.52
9.49
9.54
8.99
9.49
65. Movie Theater Revenue Box office revenues for movie theaters have remained high
for the last decade, in spite of the recent popularity of DVD players and home movie
theaters. The function R represented by the table in the margin shows the annual U.S.
box office revenue (in billions).
(a) Find R 120002 and R 120062 . What do these values represent?
(b) For what values of x is R 1x2 = 9.49?
(c) Find the net change in box office revenues from 2000 to 2006.
66. Biotechnology The biotechnology industry is responsible for hundreds of medical,
environmental, and technological advances; one such advance is DNA fingerprinting. In
the table below, the function f gives annual biotechnology revenue and the function g
gives annual research and development (R&D) revenue, in billions of dollars, between
1995 and 2005.
(a) Find f 119952 , f 120052 , g11995 2 , and g120052 . What do these values represent?
(b) Find x and y so that f 1x2 = f 1y 2 .
(c) Find the net change in biotechnology revenues and in R&D expenses from 1995
to 2005.
x
(year)
f(x)
(billions of dollars)
g(x)
(billions of dollars)
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
12.7
14.6
17.4
20.2
22.3
26.7
29.6
29.6
39.2
43.8
50.7
7.7
7.9
9.0
10.6
10.7
14.2
15.7
20.5
17.9
19.6
19.8
67. Income Tax In a certain country, income tax T is assessed according to the following
function of income x (in dollars):
0
T 1x2 = • 0.08x
1600 + 0.151x - 20,0002
if 0 … x … 10,000
if 10,000 6 x … 20,000
if 20,000 6 x
(a) Find T(5000), T(12,000), and T(25,000). What do these values represent?
(b) Find the tax on an income of $20,000.
(c) If a businessman pays $25,000 in taxes in this country, what is his income?
SECTION 1.5
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Function Notation: The Concept of Function as a Rule
63
68. Interest Income Greg earned $5000 over the summer working on a construction site. He
deposits his earnings in a money market account that offers a graduated interest rate based
on the balance of the account. If the balance is between $2500 and $20,000, the account
earns 2.5% interest per annum; if the balance is above $20,000, the account earns 3.68%;
and if the balance falls below $2500, the bank pays no interest but charges a $13 fee per
month. The interest i earned in 1 month on a balance of x dollars is modeled by the function
- 13
0.025
x
i 1x2 =
e 12
0.0368
x
12
if 0 … x 6 2500
if 2500 … x … 20,000
if 20,000 6 x
(a) Find i 110,0002 and i 120,0002 . What do these values represent?
(b) Find the interest Greg earns in the first month (on his initial $5000 deposit).
(c) Find the balance Greg needs to have in his account to earn $70 interest in one month.
69. Internet Purchases An Internet bookstore charges $15 shipping for orders under
$100 but provides free shipping for orders of $100 or more. The cost C of an order is a
function of the total price x of the books purchased.
(a) Express C as a piecewise defined function:
C1x 2 = e
___________
___________
if x 6 100
if x Ú 100
(b) Find C(75), C(100), and C(105). What do these values represent?
70. Cost of a Hotel Stay A hotel chain charges $75 each night for the first two nights and
$50 for each additional night’s stay. The total cost T is a function of the number of
nights x that a guest stays.
(a) Express T as a piecewise defined function:
T1x2 = e
___________
___________
if 0 … x … 2
if 2 6 x
(b) Find T(2), T(3), and T(5). What do these values represent?
71. Speeding Tickets In a certain state the maximum speed permitted on freeways is
65 mi/h, and the minimum is 40 mi/h. The fine for violating these limits is $15 for every
mile above the maximum or below the minimum speed. So the fine F is a function of
the driving speed x (mi/h) on the freeway.
(a) Express F as a piecewise defined function:
_________
F1x2 = c _________
_________
if 0 6 x 6 40
if 40 … x … 65
if 65 6 x
(b) Find F(30), F(50), and F(75). What do these values represent?
(c) A driver is assessed a fine of $225. What are the two possible speeds at which he
could have been driving when he was caught?
72. Utility Charges A utility company charges a base rate of 10 cents per kilowatt hour
(kWh) for the first 350 kWh and 15 cents per kilowatt hour for all additional electricity
usage. The amount E that the utility company charges is a function of the number x of
kilowatt hours used.
(a) Express E as a piecewise defined function:
E1x2 = e
___________
___________
if 0 … x … 350
if 350 6 x
(b) Find E(300), E(350), and E(600). What do these values represent?
(c) One bill for electric usage is $65.67. How many kilowatt hours are covered by this bill?