In mathematics, the inverse function of a function \(f\) (also called the inverse of \(f\)) is a function that undoes the operation of \(f\). The inverse of \(f\) exists if and only if \(f\) is bijective, and if it exists, is denoted by \(\displaystyle f^{-1}\).

For a function \(f\colon X\to Y\), its inverse \(\displaystyle f^{-1}\colon Y\to X\) admits an explicit description: it sends each element \(y\in Y\) to the unique element \(x\in X\) such that \(f(x) = y\).

(“Inverse Function” 2022)

For example, if \(f\) is the function


then to determine \(f^{-1}(y)\) for a real number \(y\), one must find the unique real number \(x\) such that \((2x + 8)3 = y\). This equation can be solved:

\(\begin{aligned}y&=(2x+8)^{3}\{\sqrt[{3}]{y}}&=2x+8\{\sqrt[{3}]{y}}-8&=2x\{\dfrac {{\sqrt[{3}]{y}}-8}{2}}&=x.\end{aligned}\)

Thus the inverse function \(f^{-1}\) is given by the formula

\(f^{-1}(y)={\frac {{\sqrt[{3}]{y}}-8}{2}}\).