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Injective function

A mathematical function is called injective (or one-to-one or an injection) if the function maps no more than one possible input value to each possible output value. (This is in contrast to a "many to one" function, which maps two or more input values to some output values).

More formally, a function fX → Y is injective if for every y in the codomain Y there is at most one x in the domain X with f(x) = y. Put another way, given x and x' in X, if f(x) = f(x'), then it follows that x = x'.

Surjective, not injective

Injective, not surjective


Not surjective, not injective

When X and Y are both the real line R, then an injective function fR → R can be visualized as one whose graph is never intersected by any horizontal line more than once. (This is the horizontal line test.)

Examples and counterexamples

Consider the function fR → R defined by f(x) = 2x + 1. This function is injective, since given arbitrary real numbers x and x', if 2x + 1 = 2x' + 1, then 2x = 2x', so x = x'.

On the other hand, the function gR → R defined by g(x) = x2 is not injective, because (for example) g(1) = 1 = g(−1).

However, if we define the function hR+ → R by the same formula as g, but with the domain restricted to only the nonnegative real numbers, then the function h is injective. This is because, given arbitrary nonnegative real numbers x and x', if x2 = x'2, then |x| = |x'|, so x = x'.


See also: Surjection, Bijection