The Inverse of a [[Function]] $f$ is a [[Function]] (denoted as $f^{-1}$) that maps every [[Image]] of $f$ to its [[Pre-Image]].
[[Composition|Composing]] $f$ with [[Domain]] $X$ and its inverse $f^{-1}$ is equal to the [[Identity|Identity Function]] of $X$.
$\huge \begin{align}
f&: X\to Y \\
f^{-1} &: Y \to X \\
f^{-1} \circ f &=\op{id}_{X} \\
f^{-1}(f(x)) &= x
\end{align}$
Any [[Function]] $f$ is invertible [[Biconditional|if and only if]] it is [[Bijective]], in other words, every element in $Y$ has a [[Pre-Image]].
If a [[Composition]] between a function $f$ and a function $g$ on the left, $f\circ g$, is equal to the [[Identity|Identity Function]], then this implies $g\circ f=\op{id}$, therefore $g$ and $f$ are each others inverse.
The [[Logical Inverse|Inverse]] of any [[Function]] $f