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.

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.)

EXAMPLE 5 Find the inverse of a cubic function Consider the function f (x) = 2x3 + 1. 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:

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.

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

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

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

GUIDED PRACTICE GUIDED PRACTICE for Examples 4 and 5 7. f(x) = – x3 64 125 ANSWER f –1(x) = – 3√ x 5 4

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

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

√ GUIDED PRACTICE GUIDED PRACTICE for Examples 4 and 5 10. g(x) = –7x5 + 7 ANSWER g –1 (x) = √ x – 7 –7 5