In mathematics, a surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there exists at least one element x in the function's domain such that f(x) = y. In other words, for a function f : X → Y, the codomain Y is the image of the function's domain X. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.

Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom of choice, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.

A surjective function is a function whose image is equal to its codomain. Equivalently, a function ${\displaystyle f}$ with domain ${\displaystyle X}$ and codomain ${\displaystyle Y}$ is surjective if for every ${\displaystyle y}$ in ${\displaystyle Y}$ there exists at least one ${\displaystyle x}$ in ${\displaystyle X}$ with ${\displaystyle f(x)=y}$. Surjections are sometimes denoted by a two-headed rightwards arrow (U+21A0 ↠ RIGHTWARDS TWO HEADED ARROW), as in ${\displaystyle f\colon X\twoheadrightarrow Y}$.

Symbolically,

A function is bijective if and only if it is both surjective and injective.

If (as is often done) a function is identified with its graph, then surjectivity is not a property of the function itself, but rather a property of the mapping. This is, the function together with its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone.

The function g : Y → X is said to be a right inverse of the function f : X → Y if f(g(y)) = y for every y in Y (g can be undone by f). In other words, g is a right inverse of f if the composition f o g of g and f in that order is the identity function on the domain Y of g. The function g need not be a complete inverse of f because the composition in the other order, g o f, may not be the identity function on the domain X of f. In other words, f can undo or "reverse" g, but cannot necessarily be reversed by it. For example, ${\displaystyle f:\mathbb {R} ^{2}\rightarrow \mathbb {R} ,(x,y)\mapsto x}$ has right inverse ${\displaystyle g:z\mapsto (z,0);}$ the composition in the other order gives ⁠${\displaystyle g(f((x,y)))=(x,0)}$⁠, which equals ⁠${\displaystyle (x,y)}$⁠ only if ⁠${\displaystyle x=0}$⁠.