surjective

Surjective describes a property of a function with respect to its codomain. A function f: X → Y is surjective (onto) if every element of Y is the image of at least one element of X. In formal terms, for every y in Y there exists an x in X with f(x) = y. A key point is that surjectivity depends on the chosen codomain; a function can be surjective for one codomain and not for another.

Examples help illustrate. The function f: R → R defined by f(x) = x^3 is surjective, since for any y there is x = ∛y with f(x) = y. The same f would not be surjective if the codomain were restricted to a subset that excludes some real numbers. Conversely, f: R → (0, ∞) given by f(x) = e^x is not surjective onto R, but it is surjective onto (0, ∞). Another simple example is f: Z → Z, f(n) = n + 1, which is surjective because every integer m has a preimage n = m − 1.

A useful characterization is that surjective maps have right inverses: if f: X → Y is surjective, there exists a function g: Y → X with f(g(y)) = y for all y in Y. Surjectivity is preserved under composition: if f: X → Y and g: Y → Z are surjective, then g ∘ f: X → Z is surjective. A function is bijective exactly when it is both injective (one-to-one) and surjective (onto).