Surjective function

A surjective function, also called onto, hits every element of its codomain.

Formally, a Function $f:X\to Y$ is surjective when $\forall y\in Y,\exists x\in X\text{ such that }f(x)=y.$ Equivalently, the image/range equals the codomain: $f(X)=Y.$

Examples

The function $f:\mathbb{R}\to \mathbb{R}$, $f(x)=x^3$, is surjective: every real number has a real cube root.

The function $g:\mathbb{R}\to \mathbb{R}$, $g(x)=x^2$, is not surjective because no input maps to a negative output. But $h:\mathbb{R}\to [0,\infty )$, $h(x)=x^2$, is surjective. The formula did not change; the codomain did.

Why it matters

Surjectivity depends on the declared codomain, so never call a function “onto” without saying onto what. It is the condition that no target value is missed.

For inverse functions, surjectivity ensures every target value has at least one preimage. Injectivity then ensures that preimage is unique. Together they give a Bijective function.

In physics, surjectivity can describe whether a control parameter can reach every desired state in a model, or whether a measurement scale covers every possible value under consideration.

Related: Injective function, Codomain and range, Mapping.