Every function has an inverse relation. If this inverse relation is also a function, then it is called an inverse function. In other words, the inverse of a function $f$ is another function $f^{\text{-}1}$ such that they undo each other.

Also, if $x$ is the input of a function $f$ and $y$ its corresponding output, then $y$ is the input of $f^{\text{-}1}$ and $x$ its corresponding output.

Example

Consider a function $f$ and its inverse $f^{\text{-}1}.$ These functions will be shown to undo each other. To do so, it needs to be proven that $f\left(f^{\text{-}1}(x)\right)=x$ and that $f^{\text{-}1}\left(f(x)\right)=x.$ To start, the first equality will be proven. First, the definition of $f$ will be used. Now, in the above equation, $\frac{x+3}{2}$ will be substituted for $f^{\text{-}1}(x).$ A similar procedure can be performed to show that $f^{\text{-}1}\left(f(x)\right)=x.$

	Definition of First Function	Substitute Second Function	Simplify
$f\left(f^{\text{-}1}(x)\right)\stackrel{?}{=}x$	$2f^{\text{-}1}(x)-3\stackrel{?}{=}x$	$2\left(\frac{x+3}{2}\right)-3\stackrel{?}{=}x$	$x=x✓$
$f^{\text{-}1}\left(f(x)\right)\stackrel{?}{=}x$	$\frac{f(x)+3}{2}\stackrel{?}{=}x$	$\frac{2x-3+3}{2}\stackrel{?}{=}x$	$x=x✓$

Therefore, $f$ and $f^{\text{-}1}$ undo each other. The graphs of these functions are each other's reflection across the line $y=x.$ This means that the points on the graph of $f^{\text{-}1}$ are the reversed points on the graph of $f.$