Inverse Functions

An inverse function "undoes" what the original function does.

Definition

If $f(a) = b$, then $f^{-1}(b) = a$

The inverse of $f$ is denoted $f^{-1}$ (NOT $\frac{1}{f}$!)

Property of Inverses

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

The composition of a function and its inverse gives you back the original input.

Finding Inverse Functions

Steps:

1. Write $y = f(x)$
2. Swap $x$ and $y$
3. Solve for $y$
4. The result is $y = f^{-1}(x)$

Example 1

Find the inverse of $f(x) = 2x + 3$

1. $y = 2x + 3$
2. $x = 2y + 3$
3. $x - 3 = 2y$
4. $y = \frac{x - 3}{2}$

So $f^{-1}(x) = \frac{x - 3}{2}$

Example 2

Find the inverse of $f(x) = x^3$

1. $y = x^3$
2. $x = y^3$
3. $y = \sqrt[3]{x}$

So $f^{-1}(x) = \sqrt[3]{x}$

Horizontal Line Test

A function has an inverse if and only if it passes the horizontal line test: no horizontal line intersects the graph more than once.

Functions that pass this test are called one-to-one functions.

Graph of Inverse

The graph of $f^{-1}$ is the reflection of $f$ over the line $y = x$.