Question 5
Let $G$ be the set of functions $f:\mathbb R\to\mathbb R$ of the form $f(x)=ax+b$, where $a$ and $b$ are real numbers and $a\ne0$. Suppose that $G$ satisfies the following conditions:
(1) If $f,g\in G$, then $g\circ f\in G$, where $(g\circ f)(x)=g[f(x)]$.
(2) If $f\in G$ and $f(x)=ax+b$, then the inverse $f^{-1}$ of $f$ belongs to $G$ ($f^{-1}(x)=(x-b)/a$).
(3) For each $f\in G$ there exists a number $x_f\in\mathbb R$ such that $f(x_f)=x_f$.
Prove that there exists a number $k\in\mathbb R$ such that $f(k)=k$ for all $f\in G$.
(1) If $f,g\in G$, then $g\circ f\in G$, where $(g\circ f)(x)=g[f(x)]$.
(2) If $f\in G$ and $f(x)=ax+b$, then the inverse $f^{-1}$ of $f$ belongs to $G$ ($f^{-1}(x)=(x-b)/a$).
(3) For each $f\in G$ there exists a number $x_f\in\mathbb R$ such that $f(x_f)=x_f$.
Prove that there exists a number $k\in\mathbb R$ such that $f(k)=k$ for all $f\in G$.