Function and their inverse

Algebra Level 5

$\large f(x)=\begin{cases} -x+1, \quad \ \ \ \ x \leq 0 \\ -(x-1)^2, \quad x \geq 1 \end{cases}$

Define the piecewise function of $f(x)$ above, find the number of solution(s) of the equation $f(x)-f^{-1}(x)=0$ .