Track / Stage 5 / 298 of 400 #898 of 2000

Problem 898

AIME late

Algebra Difficulty 5.7 Prove it

Do there exist two functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x \in \mathbb{R}$ :

$f(g(x))=x^{2} \text { and } g(f(x))=x^{3}$

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

As $g \circ f$ is bijective, $g$ is surjective and $f$ is injective. Moreover, if $x \in \mathbb{R}$ , $f(x)^{2}=f(g(f(x)))=f\left(x^{3}\right)$ and thus $f\left(x^{3}\right)=f(x)^{2}$ . Therefore, if $x \in\{-1,0,1\}$ , we have $f(x)=f\left(x^{3}\right)=f(x)^{2}$ , hence $f(x)=0$ or 1. But this contradicts the injectivity of $f$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.