# Math Insight

### Inverse function definition

An inverse function is a function that undoes the action of the another function. A function $g$ is the inverse of a function $f$ if whenever $y=f(x)$ then $x=g(y)$. In other words, applying $f$ and then $g$ is the same thing as doing nothing. We can write this in terms of the composition of $f$ and $g$ as $g(f(x))=x$.

A function $f$ has an inverse function only if for every $y$ in its range there is only one value of $x$ in its domain for which $f(x)=y$. This inverse function is unique and is frequently denoted by $f^{-1}$ and called “$f$ inverse.”

For an overview into the idea of an inverse function, see the function machine inverse.