Surjective function

A function $f:X\to Y$ is surjective when every element of its codomain occurs as a value: for each $y\in Y$ there is an $x\in X$ with $f(x)=y$. Equivalently, its image is all of $Y$.

For example, $x\mapsto x^2$ is surjective as a function $\mathbb{R}\to {\mathbb{R}}_{\ge 0}$, but not as a function $\mathbb{R}\to \mathbb{R}$, since negative values are missed. A surjective function may identify different inputs. Together with injectivity, surjectivity gives a bijection and hence an inverse function.

A surjective function, also known as a "onto" function, is a type of function in mathematics where every element in the codomain (the set of possible outputs) is mapped to by at least one element from the domain (the set of possible inputs).