Graphing a Double Absolute Value Function

Question

How would I graph the function $y=|(|x|-1)|$ in a piecewise manner?

Answer 1

Well, the abs function is a piecewise function so you only have to consider cases.

Case 1: $|x| - 1 > 0$ then $|x|>1$ so either $x>1$ or $x≤-1$ which give you the equations $y = x-1$, and $y = -x-1$

Case 2: $|x| - 1 ≤ 0$ then $|x|≤1$ so $x≤1$ and $x≥-1$ which give you the equations $y = 1-x$, and $y = x+1$

So, your final answer is
$$y = \begin{cases} -x-1, & \text{if $x≤-1$} \\ x+1, & \text{if $-1<x≤0$} \\ 1-x, & \text{if $0<x≤1$} \\ x-1, & \text{if $1<x$} \end{cases}$$

Considering the abs function is continuous, a simple check of equivalence at endpoints shows a reasonable confidence of correctness.