How to analyze absolute value functions

Question

Can someone provide a systematic way to break ||x| - 1| into its different parts without using a graphical approach? It would be greatly appreciated if the restrictions on each part of the piece wise can be shown.

Answer 1

For $$x\geq 0$$ we have $$|x-1|$$ this is $$x-1$$ if $$x\geq 1$$ and $$-(x-1)$$ if $$x<1$$ Can you finish?

Answer 2

A smart student combines analytical methods with a graphical approach ! It is straightforward to show that the function $f(x)$ is zero for $x=+1$ and $x=−1$ and equal to $+1$ for $x=-2$, $x=0$ and $x=2$. This information allows you to draw a picture of the function, which turns out to be $W$-shaped. It is then also straightforward to write down the correct expressions for $f(x)$ on each of the 4 intervals.