Knowlpedia

Inverse function

A function that undoes a bijective function

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

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

where idA\mathrm{id}_A and idB\mathrm{id}_B are .

Remarks

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

Examples
  • For f:R→Rf:\mathbb{R}\to\mathbb{R} given by f(x)=x3f(x)=x^3, the inverse function is f−1(y)=y3f^{-1}(y)=\sqrt[3]{y}.
  • For f:Z→Zf:\mathbb{Z}\to\mathbb{Z} given by f(n)=n+1f(n)=n+1, the inverse function is f−1(m)=m−1f^{-1}(m)=m-1.