# Many-to-one

A function is called many-to-one (sometimes written ‘many-one’) if some function output value corresponds to more than one input value. In symbols, the function $f$ is many-to-one if there are two distinct values $a$ and $b$ in the domain of $f$ such that $f(a)=f(b)$. This is equivalent to saying that $f$ is not one-to-one or that $f$ is not injective.

Whether or not a function is many-to-one may depend on its domain. For example, the function $f(x)=\cos x$, $x\in\mathbb{R}$ is many-to-one (not injective) because $\cos 0=\cos2\pi$, whereas $f(x)=\cos x$, $0\le x\le \pi$ is one-to-one (injective).