Do there exist two functions $f$ and $g$ from the reals to itself satisfying $f\circ g (x)=x^2 , g\circ f (x)=x^3$ for any $x\in\mathbb{R}$?
From the given equations I could get the following information:
$f$ is injective.
$g$ is surjective and an even function.
$f(x^3)=f(x)^2$ for every real number $x$.
$g(x^2)=g(x)^3$ for every real number $x$.
How these information help us to decide whether such functions exist or not?
Thank you.
No, they don't.
I have a strong feeling this is a duplicate, but can't find the original, so let's repeat it anyway.
Say, $f(0) = a$. Then $f\circ g\circ f(0) = f(0^3)=a$, but at the same time it equals $f(0)^2=a^2$. So $a$ is either $0$ or $1$.
The same reasoning applies to $f(1)$ and $f(-1)$, with the same result. So at least some of these three values must coincide, which contradicts with $f$ being injective.