Question
Let $f(x)=2x(x-2)$ where $x \in [0,2]$. Find number of solutions of $f(f(f(x)))= \cfrac{x}{2}$
My attempt
I manually found $f(f(f(x)))$, drew its graph and $\frac{x}{2}$ and found the intersection points and got the answer which is 8. But it took a lot of time and we were supposed to do this Question in 3 minutes. Is there any simpler short method?