To find the inverse of $f:{\mathbb{R}}\rightarrow{\mathbb{R}}:$ $f(x)=x^{3}$. Here $y=f(x)$ reduces to the equation $y=x^{3}$, which has a unique solution $x=y^{1/3}.$ Hence $g:{\mathbb{R}}\rightarrow{\mathbb{R}}:$ $g(y)=y^{1/3}$ is the inverse function.

The inverse function

Let $f:A\rightarrow B$ be a function, and suppose that we can solve the equation $y=f(x)$ for $x$ to obtain $x=g(y)$; then $g:B\rightarrow A$ is the inverse function of $f$, which we write as $g=f^{-1}$ (not to be confused with the reciprocal $1/{f}$.)

In calculus, we often need to change the definition of the domain and codomain so as to ensure that an inverse function exists.