Find the possible values of `sqrt(|x|-2)` (ii) `sqrt(3-|x-1|)` (iii) `sqrt(4-sqrt(x^2))`

Explanation

(i) `sqrt(|x|-2) `
We know that square roots are defined for non-negative values only.
It implies that we must have ` |x|-2 ge 0`. Thus,
`sqrt(|x|-2) ge 0`
(ii) `sqrt(3-|x-1|)` is defined when `3-|x-1| ge 0`
But the maximum value of `3-|x-1|` is 3, when `|x-1|` is 0.
Hence, for `sqrt(3-|x-1|)` to get defined, ` 0 le 3-|x-1|le 3`.
Thus,
`sqrt(3-|x-1|) in [0, sqrt(3)]`
Alternatively, ` |x-1| ge 0`
`implies -|x-1| le 0`
`implies 3-|x-1| le 3`
But for `sqrt(3-|x-1|)` to get defined, we must have
`0 le 3 -|x-1| le3`
`implies 0 le sqrt(3-|x-1|) le sqrt(3)`
(iii) `sqrt(4-sqrt(x^(2)))=sqrt(4-|x|)`
`|x| ge 0`
`implies -|x| le 0`
`implies4-|x| le 4`
But for `sqrt(4-|x|)` to get defined `0 le 4 -|x| le 4`
` :. 0 le sqrt(4-|x|) le 2`