Functional square root

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 $\R$.