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In mathematics, a surjection is a type of function with the property that all possible output values of the function are generated as values of the function as the input to the function ranges over all possible input values.

More formally, a function fX → Y is called surjective or onto or a surjection if for every y in the codomain Y there is at least one x in the domain X with f(x) = y. Put another way, the range f(X) is equal to the codomain Y.

Surjective, not injective

Injective, not surjective


Not surjective, not injective

When X and Y are both the real line R, then a surjective function fR → R can be visualized as one whose graph will be intersected by any horizontal line.

Examples and counterexamples

Consider the function fR → R defined by f(x) = 2x + 1. This function is surjective, since given an arbitrary real number y, we can solve y = 2x + 1 for x to get a solution x = (y − 1)/2.

On the other hand, the function gR → R defined by g(x) = x2 is not surjective, because (for example) there is no real number x such that x2 = -1.

However, if we define the function hR → R+ by the same formula as g, but with the codomain has been restricted to only the nonnegative real numbers, then the function h is surjective. This is because, given an arbitrary nonnegative real number y, we can solve y = x2 to get solutions x = √y and x = −√y.


See also: Injective function, Bijection