College Algebra 9th txtbk

SECTION 3–6 Inverse Functions 245
Knowledge of this symmetry property allows us to graph f 1 if the graph of f is known,
and vice versa. Figures 7(b) and 7(c) illustrate this property for the two inverse functions
we found earlier.
If a function is not one-to-one, we can usually restrict the domain of the function to
produce a new function that is one-to-one. Then we can find an inverse for the restricted
function. Suppose we start with f(x) x 2 4. Because f is not one-to-one, f 1 does not
exist [Fig. 8(a)]. But there are many ways the domain of f can be restricted to obtain a oneto-one
function. Figures 8(b) and 8(c) illustrate two such restrictions. In essence, we are
“forcing” the function to be one-to-one by throwing out a portion of the graph that would
make it fail the horizontal line test.
Z Figure 8 Restricting the domain
of a function.
5
y
y f(x)
5
y
y x
y h(x)
5
y
y x
y g 1 (x)
5
5
x
5
5
x
5
5
x
5
5
y g(x)
5
y h 1 (x)
f(x) x 2 4
f 1 does not exist
(a)
g(x) x 2 4, x 0
g 1 (x) 1x 4, x 4
(b)
h(x) x 2 4, x 0
h 1 (x) 1x 4, x 4
(c)
Recall from Theorem 3 that increasing and decreasing functions are always one-to-one.
This provides the basis for a convenient method of restricting the domain of a function:
If the domain of a function f is restricted to an interval on the x axis over
which f is increasing (or decreasing), then the new function determined by this
restriction is one-to-one and has an inverse.
We used this method to form the functions g and h in Figure 8.
EXAMPLE 5 Finding the Inverse of a Function
Find the inverse of f(x) 4x x 2 , x 2. Graph f, f 1 , and the line y x in the same
coordinate system.
SOLUTION
Step 1. Find the domain of f and verify that f is one-to-one. We are given that the domain
of f is (, 2]. The graph of y 4x x 2 is a parabola opening downward with
vertex (2, 4) (Fig. 9). The graph of f is the left side of this parabola (Fig. 10).
From the graph of f, we see that f is increasing and one-to-one on (, 2].
y
y
5 y 4x x 2
5
y f(x)
5
5
x
5
5
x
5
5
Z Figure 9 Z Figure 10