# Inverse function

Given a function $f$, the inverse of $f$ is another function $g$ with the property that $g(f(x)) = x$ for every $x$ in the domain of $f$. If it is also true that $f(g(x))=x$ for every $x$ in the domain of $g$ then the functions form a pair of inverses.

The inverse function for $f$ is denoted by $f^{-1}$. The function $f$ has to be a bijection to possess an inverse. The domain of $f^{-1}$ will be the range of $f$ and vice versa.

For example:

• If $f(x) = x+a$, $x\in\mathbb{R}$, then $f^{-1}(x) = x-a$, $x\in\mathbb{R}$.

• The function $f(x) = \cos x$, $x\in\mathbb{R}$ has no inverse, as it is not bijective. However, the function $f(x)=\cos x$, $0\le x\le\pi$ has inverse $f^{-1}(x) = \cos^{-1} x$, $-1\le x\le1$.

• If $f(x) = e^x$, $x\in\mathbb{R}$, then $f^{-1}(x) = \ln x$, $x\in\mathbb{R}$.

Note that the graph of $y=f^{-1}(x)$ is a reflection of $y=f(x)$ in the line $y=x$.