Identity Function

The identity function arises when a function f:A→B is composed with its inverse function f^-1:B→A. $$ f^{-1} \circ f(x) = x $$ In other words: $$ f^{-1}[f(x)] = x $$

It is called the identity function because it maps every element x in the domain to itself:

$$ f^{-1}[f(x)] = x $$

The reverse composition also holds:

$$ f[f^{-1}(y)] = y $$

Here is a representation of the identity function using Euler-Venn diagrams:

The identity function is called *simple* when applied to a set, where it corresponds to a bijection.

Note. The identity function is compatible with all mathematical structures (e.g., vector spaces, algebraic structures, metric spaces, topological spaces, and so on). Depending on the context, it has different specific interpretations: in an algebraic structure, it is an isomorphism; in a vector space, a linear map; in a metric space, an isometry; and in a topological space, a homeomorphism.

A practical example

Consider the function f(x) = x²:

$$ y = x^2 $$

and its inverse: $$ x = \sqrt{y} $$

The composition of the two functions gives:

$$ f^{-1}[f(x)] = \sqrt{x^2} = x $$

This is an identity function because each element x maps back to itself.

Note. The reverse composition also holds: $$ f[f^{-1}(x)] = (\sqrt{x})^2 = x $$

Example 2

Now consider the function f(x) = 2x:

$$ y = 2x $$

and its inverse: $$ x = \frac{y}{2} $$

Their composition gives:

$$ f^{-1}[f(x)] = \frac{2x}{2} = x $$

Note. The reverse composition also holds: $$ f[f^{-1}(x)] = 2 \left( \frac{x}{2} \right) = x $$

Example 3

Finally, consider the function f(x) = sin(x):

$$ y = \sin(x) $$

and its inverse: $$ x = \arcsin(y) $$

Their composition gives:

$$ f^{-1}[f(x)] = \arcsin[\sin(x)] = x $$

Note. The reverse composition also holds: $$ f[f^{-1}(x)] = \sin(\arcsin x) = x $$

And so on.