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f(x/(x^2 + x + 1))=x^2/(x^4 + x^2 + 1)

Source: Flanders Math Olympiad 2021 p4

December 24, 2022
inequalitiesfunctionalalgebrafunctional equation

Problem Statement

(a) Prove that for every x∈Rx \in R holds that −1≤xx2+x+1≤13-1 \le \frac{x}{x^2 + x + 1} \le \frac 13
(b) Determine all functions f:R→Rf : R \to R for which for every x∈Rx \in R holds that f(xx2+x+1)=x2x4+x2+1f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}
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