My question is, why doesn't $g(f(g^{-1}(x)))=x$ imply $g(f(x))=g(x)$. 

Suppose for example $g(x)=\frac{1}{x}$ and $f(x)=2$. We would like to find the fixed points of $g(f(g^{-1}(x)))$ i.e $g(f(g^{-1}(x)))=x$ which is easy to see $\Leftrightarrow \frac{1}{2}=x$.

On the other hand, starting from $g(f(g^{-1}(x)))=x \Rightarrow g(f(x))=g(x) \Leftrightarrow \frac{1}{2} = \frac{1}{x} \Rightarrow x=2$. which is obviously a different answear.

Thank you!