Inverse function

A function that undoes a bijective function

Inverse function

An inverse function is a function that undoes a bijection: if $f:A\to B$ is a bijective function , then its inverse function $f^{-1}:B\to A$ is defined by the rule “ $f^{-1}(b)$ is the unique $a\in A$ such that $f(a)=b$ ”. Equivalently,

f^{-1}\circ f=\mathrm{id}_A\quad\text{and}\quad f\circ f^{-1}=\mathrm{id}_B,

where $\mathrm{id}_A$ and $\mathrm{id}_B$ are identity functions .

The notation $f^{-1}$ is also used for the preimage of a subset under a function, but that operation is defined even when $f$ is not bijective. Inverse functions are best understood via composition and the identity functions they produce.

Examples:

For $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^3$ , the inverse function is $f^{-1}(y)=\sqrt[3]{y}$ .
For $f:\mathbb{Z}\to\mathbb{Z}$ given by $f(n)=n+1$ , the inverse function is $f^{-1}(m)=m-1$ .