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

Explanation

`sqrt(|x|-2)`
we know that square roots are defined for non- negative values only .
It implies that we must have `|x|-2 le 0 ` Thus
`sqrt(|x|-2) ge 0 `
(ii) `sqrt(3-|x-1|)` is defined when `3-|x-1| le 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`
`rArr -|x-1| le 0 `
`rArr 3-|x-1|le3`
But for `sqrt(3-|x-1|)` to get defined ,we must have `0 le 3 -|x-1| le 3 `
`rArr 0 le sqrt(3-|x-1| le sqrt(3)`
(iii) `sqrt(4-sqrt(x^2))=sqrt(4-|x|)`
`|x| ge 0 `
`rArr - |x| le 0 `
`rArr 4-|x| le 4 `
But for `sqrt(4-|x| )` to get defined `0 le 4 - |x| le 4 `
`therefore 0 le sqrt(4-|x|) le 2 `