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<1\end{cases}`

`|x+1|=\begin{cases}x+1&\text{for }x\ge-1\\-(x+1)&\text{for }x<-1\end{cases}`

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

You have


`f(x)=\begin{cases}1-(-(x-1))+(-(x+1))=-1&\text{for }x<-1\\1-(-(x-1))+(x+1)=2x+1&\text{for }-1\le x<1\\1-(x-1)+(x+1)=3&\text{for }x\ge1\end{cases}`