Inverse function

The inverse function of a function $f: D \to C$ is a function $g:C \to D$ that does the opposite of $f$ . A function has an inverse if and only if it is bijective. The inverse of a function $f$ is denoted by $f^{-1}$ (not to be confused with the reciprocal of $f$).

Given any two functions, $f: D \to C$ and $g:C \to D$ (notice the reversal of the domain and codomain), we say that $f$ and $g$ are inverses of each other, denoted $f=g^{-1}$ and $g=f^{-1}$ if:

• $f(g(x))=x$ for all $x \in C$
• $g(f(x))=x$ for all $x \in D$

A function that is not bijective can be "made" invertible by restricting the domain to that where the function is one-to-one and then restricting the codomain to its image on the domain restriction. For instance, the function $f:\mathbb{R} \to \mathbb{R}$ defined by $f(x)=x^2$ is not bijective, and thus has no inverse, but restricting the domain of $f$ to the interval $[0,\infty)$ , we can obtain a function $f\mid_{[0,\infty)}:[0,\infty)\to[0,\infty)$ defined by $f\mid_{[0,\infty)}(x=x^2$ , which is a bijective function.