Problem 48 *

Let f : \mathbb R \to \mathbb R, w : \mathbb R \to \mathbb R. Prove that there exist no f, w such that f\left(w(x)\right) = x^2 and w\left(f(x)\right) = x^3, but there exists f, w satisfying f \left(w(x)\right) = x^2 and w \left(f(x)\right) = x^4.

[originally posted by Lord of the Flies on TSR]