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Identity Function

Definition

Let XX be a set. The identity function on XX is the function id⁡X ⁣:X→X\operatorname{id}_X\colon X\to X defined by id⁡X(x)=x\operatorname{id}_X(x)=x for all x∈X.x\in X\text{.} That is, a function that always returns its input (argument) unchanged.

Interpretation

The identity function is the function that does nothing: it returns every input unchanged. It matters because it plays the role for composition that the number 11 plays for multiplication.

The line y = x, the graph of the identity function, with the point (a, a) marked.

The graph of the identity on R\mathbb{R} is the diagonal line y=xy=x: the marked point (a,a)(a,a) has equal coordinates. Reflecting the graph of a bijection in this line gives the graph of its inverse.

For f ⁣:X→Yf\colon X\to Y, composing with the identity on either side changes nothing: f∘id⁡X=ff\circ\operatorname{id}_X=f and id⁡Y∘f=f\operatorname{id}_Y\circ f=f. This is why inverses are described by f−1∘f=id⁡Xf^{-1}\circ f=\operatorname{id}_X and f∘f−1=id⁡Yf\circ f^{-1}=\operatorname{id}_Y; the subscript matters, since for X≠YX\neq Y these are two different functions. In Math II the identity returns as the identity matrix, which leaves every vector unchanged.

Key Results

  • Composing any function with the identity function leaves it unchanged.
  • The identity function is bijective, and it is its own inverse.

Examples

  • For any x∈X,x\in X\text{,} id⁡X(x)=x.\operatorname{id}_X(x)=x\text{.}
  • For a bijection f ⁣:X→Y,f\colon X\to Y\text{,} we have (f∘f−1)(x)=id⁡Y(x)(f\circ f^{-1})(x)=\operatorname{id}_Y(x) and (f−1∘f)(x)=id⁡X(x).(f^{-1}\circ f)(x)=\operatorname{id}_X(x)\text{.}

Prerequisites

Sources

Aliases: identity map, identity mapping Symbols: element of, real numbers

Last updated: 20 October 2026 03:02