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No functions

Source: Rioplatense Olympiad

March 4, 2018
functionalgebraalgebra unsolved

Problem Statement

Let g(x)=ax2+bx+cg(x) = ax^2 + bx + c a quadratic function with real coefficients such that the equation g(g(x))=xg(g(x)) = x has four distinct real roots. Prove that there isn't a function ff: R−−RR--R such that f(f(x))=g(x)f(f(x)) = g(x) for all xx real
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