Surjective function

In mathematics, a surjective function or onto function or surjection is a function for which every possible output value occurs for one or more input values: that is, its image is the whole of its codomain.

An surjective function f has an inverse $f^{-1}$ (this requires us to assume the Axion of Choice). If y is an element of the image set of f, then there is at least one input x such that $f(x)=y$. We define $f^{-1}(y)$ to be one of these x values. We have $f(f^{-1}(y)=y$ for all y in the codomain.

See also

• Bijective function
• Injective function