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