MathDB
f (-x) = -f (x) if f (g (x)) = g (f (x)) = x, f (x) + g (x) = x

Source: 2002 Moldova JBMO TST p12

February 25, 2021
algebrafunctions

Problem Statement

Let MM be an empty set of real numbers. For any x∈Mx \in M the functions f:M→Mf: M\to M and g:M→Mg: M\to M satisfy the relations f(g(x))=g(f(x))=xf (g (x)) = g (f (x)) = x and f(x)+g(x)=xf (x) + g (x) = x. Show that −x∈M- x \in M ¸ and f(−x)=−f(x)f (-x) = -f (x) whatever x∈Mx \in M.
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