A function for which more than one element of the domain maps onto an element in the range. So f(x) = x² is many‐to‐one if x is defined on the real numbers since f(a) = f(−a). A many‐to‐one function is not invertible, since there is no way to define a function to return more than one value. However, a many‐to‐one function can be restricted to a domain within which it is a one‐to‐one function, and for which the inverse function exists. If f(x) = x² is restricted to x ≥ 0 then $f^{-1}(x)=\sqrt{x}$ exists. See principal values.