functional equations - Existence of two functions $f$ and $g$ for which $fcirc g (x)=x^2 , gcirc f (x)=x^3$

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:

1. $f$ is injective.




2. $g$ is surjective and an even function.





3. $f(x^3)=f(x)^2$ for every real number $x$.




4. $g(x^2)=g(x)^3$ for every real number $x$.

How these information help us to decide whether such functions exist or not?

Thank you.

Answer

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.