If there exists functions $f, g$ such that $f(g(x))=x$ can we say that $g(f(x))=x)$ as well? If there exists functions $f, g$ such that $$f(g(x))=x$$ can we say that $g(f(x))=x$ as well? I don't know but it apparently appears to hold true. 
Just to clarify, my doubt lies in the fact whether it ALWAYS HOLDS or not.
Assume that $x\in \mathbb{R}$.
I am not specifying the domain and codomain of the Functions anymore as that may lead to some case wise discussion of the particular matter.

Let $g:\Bbb R \to \Bbb R^2$ be the inclusion map and $f : \Bbb R^2 \to \Bbb R$ be the projection map. Then, $f(g(x)) = f(x,0) = x$, while $g(f(x,y)) = g(x) = (x,0) \ne (x,y)$.
