Given by a graph is the function $f(x)$. How many solutions does $f(f(f(x)))=0$ have? This problem, like the other ones I've posted so far, is from a 2017 olympiad. It goes like this:
Given by the following graph is the function $f(x)$:

How many solutions does $f(f(f(x)))=0$ have?
In other words, how many time does the graph of $f(f(f(x)))$ touch or intersect the x-axis?
For me, the issue already lies in identifying $f(x)$. I know it must have something to do with $|x|$. How would one go about solving this problem?

You don't really need to obtain a different representation of $f$ besides the graph, to solve this problem. 
If $f^3(x)=0$, then $f^2(x)$ must be either $0$ or $2$. 
So, the problem becomes computing how many solutions are there of $f^2(x)=0$ union the solutions of $f^2(x)=2$. 
For $f^2(x)=0$ we have again $f(x)=0$ or $2$. Therefore, $x=0,2$ or $-2$.
For $f^2(x)=2$ we have $f(x)=-2$, from where $x=4$.
So, in total we have $4$ solutions $x=0,x=2, x=-2$ and $x=4$.
