Let $f:[0,1]\to \mathbb{R},f(x)=4x(1-x),f_n(x)=f(f_{n-1}(x))\;\forall n\geq 1$ and $f_0(x)=x$. Find number of solutions to the equation $f_n(x)=x$.

Question

Let $f:[0,1]\to \mathbb{R},f(x)=4x(1-x),f_n(x)=f(f_{n-1}(x))\;\forall n\geq 1$ and $f_0(x)=x$.

Find number of solutions to the equation $f_n(x)=x$.

My Attempt

I tried plotting graphs of $y=f(x)$ and $y=f(f(x))$ by focusing on the fact when $f(x)=0$ and when $f(x)=1$ because $f(f(x))=0$ at these points and also found the points when $f(f(x))=1$. So I got that $f(f(x))=x$ has $3$ solutions. Can there be a recursive solution or some other approach here because this way I am not able to get any pattern.