Inverse functions and $f(x)=x$

Question

Is every solution of $f(x)=x$ a solution of $f^{-1}(x) = f(x)$?

If not, why not?

Can we not do the following ? \begin{eqnarray} f(x)&=&x\\ f(f(x))&=&f(x)=x\\ f^{-1}(x)&=&f(x)\\ \end{eqnarray}

Also, what about the converse of this statement?

What if a function is invertible (bijective) only in a certain domain?