Lesson 16.4: Inverse Functions

Property

Two functions are inverse functions if each one undoes the effect of the other. If $f$ and $g$ are inverse functions, then applying $f$ followed by $g$ (or $g$ followed by $f$) returns you to the original input value. We can find an inverse by interchanging the variables in the function. For example, $y = \sqrt[3]{x}$ if and only if $x = y^3$.

Examples

Property

If $f(x)$ is a one-to-one function whose ordered pairs are of the form $(x, y)$, then its inverse function $f^{-1}(x)$ is the set of ordered pairs $(y, x)$.
The domain of $f$ is the range of $f^{-1}$ and the domain of $f^{-1}$ is the range of $f$.
The graphs of $f$ and $f^{-1}$ are mirror images of each other through the line $y=x$.

Examples

• The inverse of the function {(0, 1), (2, 5), (4, 9)} is the function {(1, 0), (5, 2), (9, 4)}.

• If the function $f$ has domain {0, 2, 4} and range {1, 5, 9}, its inverse $f^{-1}$ has domain {1, 5, 9} and range {0, 2, 4}.

Property

A function does not have an inverse if it fails the horizontal line test, meaning there exists at least one horizontal line that intersects the graph at two or more points. This occurs when $f(a) = f(b)$ for some $a \neq b$ in the domain.

Examples