Let $f(x) = |x − 2|.$ Sketch $y = f(f(f(|x|)))$ for $x\in \mathbb{R}.$

Why not start by sketching $f(x)?$

How does changing the argument to $|x|$ affect the graph? What would $f(|x|)$ look like?

What about $f(f(|x|))?$

A good way to think about this graph is to consider the sequence of transformations we are performing. Applying $|\cdot|$ to the argument of the function makes it symmetrical about the y-axis. Subtracting $2$ shifts the entire graph downwards by $2$ units and applying $|\cdot|$ to the result flips parts of the graph below the x-axis. By sketching the simpler cases, we can deduce the pattern that $n$ applications of $f$ results in $n$ peaks.