Function contradiction

Question

Let $f(x)$ and $g(x)$ be two functions and $f(x),g(x):\mathbb{R}\rightarrow\mathbb{R}$, such that $$f(f(...f(x)))=g(x)$$ thus $f(g(x))=g(x)\Rightarrow f(x)=x\Rightarrow f(f(...f(x)))=x$, what's wrong if $g(x)\neq x$ ?

Answer 1

The iteration sequence $x, f(x), f(f(x)), ...$, if it converges, will always converge to a fixed (or stationary) point $x_0$ (that is, $f(x_0) = x_0$), as you proved.

That said, $g(x)$ will be identity only on those stationary points. Otherwise it will be equal to the stationary point you reach starting $x$. Thus, the range of $g$ will be the set of stationary points. As an example (like Conrad mentioned), $f(x)=\frac{x}{2}$ and $g(x)=0$ works.