128
◆
Chapter 4
Functions
Find f (115), f (155), and f (215).
52. The power, P, in watts, in a resistance, R, in which flows a current, I, is given by
P f (I) I 2R
If R 25.6 , find f (1.59), f (2.37), and f (3.17).
53. The sound intensity of the siren mounted on a fire truck can be expressed as a
function of the distance (in kilometres) from the siren: f (x) 12 0.2x 2. Find: f (3.7),
f (6.5)
54. The temperature in degrees Celsius of a residential fire as a function of distance in metres
is f (x) 800 0.73x 2. Find the temperatures f (19) and f (29).
55. The distance d (in metres) at which smoke of height h (in metres) from a fire can be seen
is specified in the function d(h) 2.8h2 25. Evaluate d(16).
4–3
Composite Functions and Inverse Functions
Composite Functions
Just as we can substitute a constant or a variable into a given function, we can substitute a function into a function. This is known as a composite function.
◆◆◆
Example 40: If g(x) x 1, find
(a) g(2)
(b) g(z2)
(c) g[ f (x)]
(b) g(z2) z2 1
(c) g[ f (x)] f (x) 1
Solution:
(a) g(2) 2 1 3
In calculus, we will take
derivatives of composite
functions by means of the
chain rule.
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The function g[ f (x)] (which we read “g of f of x”), being made up of the two functions g(x)
and f (x), is called a composite function. If we think of a function as a machine, it is as if we
are using the output f (x) of the function machine f as the input of a second function machine g.
x S f S f (x) S g S g[ f (x)]
We thus obtain g[ f (x)] by replacing x in g(x) by the function f (x).
Example 41: Given the functions g(x) x 1 and f (x) x3, write the composite
function g[ f (x)].
◆◆◆
Solution: In the function g(x), we replace x by f (x).
g(x) x 1
g[ f (x)] f (x) 1
x3 1
since f (x) = x3.
As we have said, the notation g[ f (x)] means to substitute f (x) into the function g(x). On the
other hand, the notation f [g(x)] means to substitute g(x) into f (x).
x S g S g(x) S f S f [g(x)]
In general, f [g(x)] will not be the same as g[ f (x)].
Section 4–3
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◆
129
Composite Functions and Inverse Functions
Example 42: Given g(x) x2 and f (x) x 1, find the following:
(a) f [g(x)]
(c) f [g(2)]
(b) g[ f (x)]
(d) g[ f (2)]
Solution
f [g(x)] g(x) 1 x2 1
g[ f (x)] [ f (x)]2 (x 1)2
f [g(2)] 22 1 5
g[ f (2)] (2 1)2 9
(a)
(b)
(c)
(d)
Notice that here f [g(x)] is not equal to g[ f (x)].
Inverse of a Function
Consider a function f that, given a value of x, returns some value of y.
xS f Sy
If that y is now put into a function g that reverses the operations in f so that its output is the
original x, then g is called the inverse of f.
xS f SyS g Sx
The inverse of a function f (x) is often designated by f 1(x).
x S f S y S f 1 S x
Common
Error
◆◆◆
1
Do not confuse f 1 with .
f
Example 43: Two such inverse operations are “cube” and “cube root.”
x S cube x S x3 S take cube root S x
Thus if a function f (x) has an inverse f 1(x) that reverses the operations in f (x), then the
composite of f (x) and f 1(x) should have no overall effect. If the input is x, then the output must
also be x. In symbols, if f (x) and f 1(x) are inverse functions, then
f 1[ f (x)] x
and
f [ f 1(x)] x
Similarly, if g[ f (x)] x and f [g(x)] x, then f (x) and g(x) are inverse functions.
◆◆◆
Example 44: Using the example of the cube and cube root, if
f (x) x3
and
3 g(x) 兹
x
then
3 3
3
g[ f (x)] 兹
f (x) 兹
x x
and
3 3
f [g(x)] [g(x)]3 (兹
x) x
This shows that f (x) and g(x) are indeed inverse functions.
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130
Chapter 4
◆
Functions
To find the inverse of a function y f (x):
1. Solve the given equation for x.
2. Interchange x and y.
Example 45: We use the cube and cube root example one more time. Find the inverse g(x)
of the function
◆◆◆
y f (x) x3
Solution: We solve for x and get
3 x 兹
y
It is then customary to interchange variables so that the dependent variable is y. This gives
3 y = f 1(x) 兹
x
3 Thus f 1(x) 兹
x , is the inverse of f (x) x3, as verified earlier.
◆◆◆
◆◆◆
Example 46: Find the inverse f 1(x) of the function
y f (x) 2x 5
Solution: Solving for x gives
y5
x = 2
Interchanging x and y, we obtain
x5
◆◆◆
y f 1(x) 2
Sometimes the inverse of a function will not be a function itself, but it may be a relation.
◆◆◆
Example 47: Find the inverse of the function y x2.
Solution: Take the square root of both sides.
x 兹
y
Interchanging x and y, we get
y 兹
x
In this example, a single value of x (say, 4) gives two values of y (2 and 2), so the inverse
◆◆◆
does not meet the definition of a function. It is, however, a relation.
We cover the inverse
trigonometric functions in Secs.
7–3, 15–1, and 16–1, and
graph them in Sec. 18–6. The
exponential function and its
inverse, the logarithmic function,
are covered in Chapter 20.
Sometimes the inverse of a function gets a special name. The inverse of the sine function,
for example, is called the arcsin, and the inverse of an exponential function is a logarithmic
function.
Exercise 3
◆
Composite Functions and Inverse Functions
Composite Functions
1. Given the functions g(x) 2x 3 and f (x) x2, write the composite function g[ f (x)].
2. Given the functions g(x) x2 1 and f (x) 3 x, write the composite function f [g(x)].
131
Review Problems
3. Given the functions g(x) 1 3x and f (x) 2x, write the composite function g[ f (x)].
4. Given the functions g(x) x 4 and f (x) x2, write the composite function f [g(x)].
Given g(x) x3 and f (x) 4 3x, find:
5. f [g(x)]
6. g[ f (x)]
7. f [g(3)]
8. g[ f (3)]
Inverse Functions
Find the inverse of:
9. y 8 3x
10. y 5(2x 3) 4x
11. y 7x 2(3 x)
12. y (1 2x) 2(3x 1)
13. y 3x (4x 3)
14. y 2(4x 3) 3x
15. y 4x 2(5 x)
16. y 3(x – 2) – 4(x 3)
Case Study Discussion—Synchronizing Special Effects
Using the formula t(d ) 兹2d / g (a function), where gravity g 9.8 m/s2, you are able to
calculate the time (t) of the fall as a function of the distance (d ) it drops. If you calculate
for 4 m, 5 m, and 6 m drops you could use function notation to show that you are calculating time for specific distances, like this:
Since t(d ) 兹 2d / g
t(4) 兹 2(4) / 9.8 0.904 s
t(5) 兹 2(5) / 9.8 1.010 s
t(6) 兹 2(6) / 9.8 1.107 s
So, in this case, you have about a one second drop with only about a tenth of a second
difference for a one metre change in distance.
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CHAPTER 4 REVIEW PROBLEMS
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1. Which of the following relations are also functions?
(a) y 5x3 2x2
(b) x2 y2 25
(c) y 兹 2x
2. Write y as a function of x if y is equal to half the cube of x, diminished by twice x.
3. Write an equation to express the surface area S of a sphere as a function of its radius r.
4. Find the domain and range for each function.
5
1x
(a) y (b) y 1x
兹3 x