nonsurjective

Nonsurjective is an adjective used to describe a function that is not surjective (not onto). For a function f: A → B to be surjective, every element of the codomain B must be the value of f at some input from A. A function is nonsurjective when this condition fails: there exists at least one y in B such that no element x in A satisfies f(x) = y. Equivalently, the image f(A) is a proper subset of B.

The concept of surjectivity depends on the chosen codomain. A function can be surjective onto one codomain and not surjective onto another. For example, the map f: {1, 2} → {a, b} with f(1) = a and f(2) = b is surjective (indeed bijective). If the codomain is enlarged to {a, b, c}, the same function becomes nonsurjective because c is not hit by any input.

Examples illustrate nonsurjectivity. The function f: Z → Z given by f(n) = n^2 is nonsurjective since negative integers are not in the image. The exponential function f: R → R with f(x) = e^x is not surjective onto R because its range is (0, ∞). However, f: R → (0, ∞), f(x) = e^x, is surjective.

Related facts: a function has a right inverse if and only if it is surjective; therefore nonsurjective functions do not have a right inverse. A function can be nonsurjective, injective, both, or neither.