Question

Consider the function f(x)= 1-|x-1|+|x+1| for -3<x<3. Recall that the absolute function can be represented as a piece wise function: $|x| = \left \{ {{-x ,x\ \textless \ 0} \atop {x, x \geq 0}} \right.$ Use this to express f(x) as a piece wise function.
hence graph.

Answer 1

By the definition for the absolute value,

$|x-1|=\begin{cases}x-1&\text{for }x\ge1\\-(x-1)&\text{for }x$
$|x+1|=\begin{cases}x+1&\text{for }x\ge-1\\-(x+1)&\text{for }x$

So for the compound function $f(x)=1-|x-1|+|x+1|$, there are three intervals to consider. What happens when $x$? when $-1\le x$? when $x\ge1$?

You have

$f(x)=\begin{cases}1-(-(x-1))+(-(x+1))=-1&\text{for }x$