Identity function

Short description: In mathematics, a function that always returns the same value that was used as its argument

thumb|[[Graph of a function|Graph of the identity function on the real numbers]]

In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality f(x) = x is true for all values of x to which f can be applied.

Definition

Formally, if X is a set, the identity function f on X is defined to be a function with X as its domain and codomain, satisfying

f(x) = x   for all elements x in X.[1]

In other words, the function value f(x) in the codomain X is always the same as the input element x in the domain X. The identity function on X is clearly an injective function as well as a surjective function, so it is bijective.[2]

The identity function f on X is often denoted by idX.

In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of X.[3]

Algebraic properties

If f : XY is any function, then we have f ∘ idX = f = idYf (where "∘" denotes function composition). In particular, idX is the identity element of the monoid of all functions from X to X (under function composition).

Since the identity element of a monoid is unique,[4] one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.