Question

What is the range of `-abs(x-1)+2`?

Answer 1

We first look at `|x-1|` which is allways `>=0`
So the negative is allways `<=0`

Explanation:

`->-|x-1|<=0->-|x-1|+2<=2`

Since there is no upper limit to `|x-1|`, there is no lower limit to the equation as a whole, or:
`- oo <-|x-1|+2<=2`
graph{-|x-1|+2 [-14.24, 14.24, -7.12, 7.12]}