---
tag: H27ZC95KN5
kind: definition
name: Functional square root
msc: [39B12, 26A18]
defines: E7890PHMGP
uses: [0EPAT51DXQ, 27C6J3YNZY]
sources:
  - url: https://en.wikipedia.org/wiki/Functional_square_root
---

## Definition

Let $X$ be a set and let $f \colon X \to X$ be a [[function|0EPAT51DXQ]].
A *functional square root* of $f$ is a function $g \colon X \to X$ such that
$$g(g(x)) = f(x) \quad \text{for all } x \in X.$$
So it is an [[iterative root|27C6J3YNZY]] of order $2$ of $f$.

## Remarks

Example: on $X = (0, \infty)$, let $f(x) = \frac{x}{1 + 2x}$ and $g(x) = \frac{x}{1 + x}$.
Check: $g(g(x)) = \frac{x/(1+x)}{1 + x/(1+x)} = \frac{x}{1 + 2x} = f(x)$.

Non-example: $g(x) = \frac{x}{2}$ is not a functional square root of $f(x) = \frac{x}{2}$, since $g(g(x)) = \frac{x}{4} \ne \frac{x}{2}$ for $x \ne 0$.

Common notations are $g = f^{1/2}$ and $g = f^{[1/2]}$; the notation hides the fact that roots are not unique.
For instance $x \mapsto \pi - x$ and $x \mapsto x$ are both functional square roots of the identity of $\R$.