College Algebra 9th txtbk

MATCHED PROBLEM 2
SECTION 3–6 Inverse Functions 241
Use Theorem 5 to decide if these two functions are inverses.
f(x) 2
5 (11 x)
g(x) 5 2 x 11
There is one obvious question that remains: when a function is defined by an equation, how
can we find the inverse? Given a function y f(x) , the first coordinates of points on the graph
are represented by x, and the second coordinates are represented by y. Finding the inverse by
reversing the order of the coordinates would then correspond to switching the variables x and
y. This leads us to the following procedure, which can be applied whenever it is possible to
solve y f(x) for x in terms of y.
Z FINDING THE INVERSE OF A FUNCTION f
Step 1. Find the domain of f and verify that f is one-to-one. If f is not one-to-one,
then stop, because f 1 does not exist.
Step 2. If the function is written with function notation, like f(x), replace the function
symbol with the letter y. Then interchange x and y.
Step 3. Solve the resulting equation for y. The result is f 1 (x).
Step 4. Find the domain of f 1 . Remember, the domain of f 1 must be the same
as the range of f.
You can check your work using Theorem 5.
EXAMPLE 3 Finding the Inverse of a Function
Find f 1 for f(x) 1x 1.
y
5
y f(x)
5
5
f(x) x 1, x 1
Z Figure 4
SOLUTION
x
Step 1. Find the domain of f and verify that f is one-to-one. Since 1x 1 is defined
only for x 1 0, the domain of f is [1, ). The graph of f in Figure 4 shows
that f is one-to-one, so exists.
f 1
Step 2. Replace f(x) with y, then interchange x and y.
y 1x 1
x 1y 1
Step 3. Solve the equation for y.
x 1y 1
x 2 y 1
x 2 1 y
The inverse is f 1 (x) x 2 1.
Step 4. Find the domain of f 1 .
Interchange x and y.
Square both sides.
Add 1 to each side.
The equation we found for is defined for all x, but the domain should be the range of f.
From Figure 4, we see that the range of f is [0, ) so that is the domain of f 1 .
Therefore,
f 1
f 1 (x) x 2 1 x 0