Problem

Source: Morocco TST 2010, problem 7

Tags: function, algebra



Let $G$ be a non-empty set of non-constant functions $f$ such that $f(x)=ax + b$ (where $a$ and $b$ are two reals) and satisfying the following conditions: 1) if $f \in G$ and $g \in G$ then $gof \in G$, 2) if $f \in G$ then $f^ {-1} \in G$, 3) for all $f \in G$ there exists $x_f \in \mathbb{R}$ such that $f(x_f)=x_f$. Prove that there is a real $k$ such that for all $f \in G$ we have $f(k)=k$