Inverse (function)

Bijections always come in pairs, if $f:A\to B$ is a bijection, then there is a function $\boxed{f^{-1}:B\to A}$, it is called the inverse of $f$, defined by:

$$f^{-1}(b)=a\text{ whenever }f(a)=b$$

This also means $f^{-1}$ is also a bijection, so we have:

$$\begin{array}{rl}{f}^{-1}(f(a))& =a\text{ for every }a\in A\\ f^{-1}(f(b))& =b\text{ for every }b\in B\end{array}$$

For example, let Odd and Even be the sets of odd and even natural numbers, respectively. Define a function $f:\text{Odd }\to \text{ Even}$ by $f(n)=n-1$. Then $f$ is a bijection and its inverse $f^{-1}:\text{Even }\to \text{ Odd}$ is defined as $f^{-1}(n)=n+1$.