When solving functional equations it can be helpful to substitute another function, say, g(x) rather than x to the original functional equation h(x). Under what condition is this a permissible substitution that leaves the original solution intact? Is bijectivity sufficient and if so, why is it?
Example: h(1-h(x))=x. Can I substitute g(x) for x in this equation?
Any substitution is possible whenever the necessary assumptions are fulfilled. They are often helpful to simplify the equation in question. Let $h\colon\Bbb R\to\Bbb R$. Put $1-h(x)$ instead of $x$. Hence $h(1-x)=1-h(x)$, so $$h(x)+h(1-x)=1,$$ which, maybe, maynot, allows us to gain some extra properties of the solution. For instance, for $x=\frac{1}{2}$ we get $h\bigl(\frac{1}{2}\bigr)=\frac{1}{2}.$ This is not so visible by the original equation.