Inverse Function Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

f(x) = 2^x

, find

f^{-1}(x)

. Then verify by showing that

f(f^{-1}(8)) = 8

1
Set $y = 2^x$ . To find the inverse, swap $x$ and $y$ : $x = 2^y$ . Solve for $y$ : $y = \log_2 x$ .
2
So $f^{-1}(x) = \log_2 x$ .
3
Verify: $f^{-1}(8) = \log_2 8 = 3$ . Then $f(3) = 2^3 = 8$ . ✓

Answer

f^{-1}(x) = \log_2 x

Exponential and logarithmic functions are inverses of each other. The composition

f(f^{-1}(x)) = x

always holds for inverse functions — this is the defining property.

About Inverse Function

The inverse of a function $f$ is a function $f^{-1}$ that reverses $f$ : if $f(a) = b$ then $f^{-1}(b) = a$ . It exists only when $f$ is one-to-one.

Find the inverse of [formula].

Find the inverse of [formula] for [formula].

Find the inverse of [formula].