Using the definition, prove that the function $f:A \to B$ is invertible if and only if $f$ is both one-one and onto.

A function $f:X \to Y$ is defined to be

invertible, if there exist a function $g = Y \to X$ such that $gof = {I_X}$ and $fog = {I_Y}$.

The function is called the inverse of $f$ and is denoted by ${f^{ - 1}}$.

A function $f = X \to Y$ is invertible iff $f$ is a bijective function.