Question 5
Let $R$ be the set of real numbers. Show that there are no functions $f, g: R \to R$ such that $g(f(x)) = x^3$ and $f(g(x)) = x^2$ for all $x$. Let $S$ be the set of all real numbers greater than $1$. Show that there are functions $f, g: S \to S$ satisfying the condition above.