I was reading "Functional Equations and How to Solve Them" by Small and the following comment pops up without much justification on p. 13:
If $a(x)$ is an involution, then $f(a(x))=f(x)$ has as solutions $f(x) = T\,[x,a(x)]$, where $T$ is an arbitrary symmetric function of $u$ and $v$.
I was wondering why this was true (it works for examples I've tried, but I am not sure $(1)$ how to prove this and $(2)$ if there's anything obvious staring at me in the face here).