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).