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 \(f(x) = y\).
For example, if \(f\) is the function
\(f(x)=(2x+8)^{3}\)
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
\(f^{-1}(y)={\frac {{\sqrt[{3}]{y}}-8}{2}}\).