1. Find the inverse of a power function
EXAMPLE 4
Find the inverse of a power function
Find the inverse of f(x) = x2, x ≥ 0. Then graph f and f –1.
SOLUTION
f(x) = x2
Write original function.
y = x2
Replace f (x) with y.
x = y2
Switch x and y. = x y
Take square roots of each side.

2. EXAMPLE 4
Find the inverse of a power function
The domain of f is restricted to nonnegative values of x. So, the range of f –1 must also be restricted to nonnegative values, and therefore the inverse is f –1(x) = x. (If the domain was restricted to x ≤ 0, you would choose f –1 (x) = – x.)

3. EXAMPLE 5
Find the inverse of a cubic function
Consider the function f (x) = 2x Determine whether the inverse of f is a function. Then find the inverse.
SOLUTION
Graph the function f. Notice that no horizontal line intersects the graph more than once. So, the inverse of f is itself a function. To find an equation for f –1, complete the following steps:

4. Find the inverse of a cubic function
EXAMPLE 5
Find the inverse of a cubic function
f (x) = 2x3 + 1
Write original function.
y = 2x3 + 1
Replace f (x) with y.
x = 2y3 + 1
Switch x and y.
x – 1 = 2y3
Subtract 1 from each side.
x – 1 2
= y3
Divide each side by 2. 3
x – 1 2
= y
Take cube root of each side.

5. EXAMPLE 5
Find the inverse of a cubic function
ANSWER
The inverse of f is f –1(x) = 3
x – 1 2 .

6. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
Find the inverse of the function. Then graph the function and its inverse.
5. f(x) = x6, x ≥ 0
ANSWER
f –1(x) = 6√ x

7. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
6. g(x) = x3 1 27
ANSWER
g–1(x) = 33√ x

8. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
7. f(x) = – x3 64
125
ANSWER
f –1(x) = – √ x 5 4

9. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 4 and 5
8. f(x) = –x3 + 4
ANSWER
f –1(x) = 3√ 4 – x

10. √ GUIDED PRACTICE GUIDED PRACTICE for Examples 4 and 5
9. f(x) = 2x5 + 3
f –1(x) = √
x – 3 2
ANSWER 5

11. √ GUIDED PRACTICE GUIDED PRACTICE for Examples 4 and 5
g(x) = –7x5 + 7
ANSWER
g –1 (x) = √
x – 7 –7 5