The domain of the function `f(x)=sqrt(1n_((|x|-1))(x^2+4x+4))` is `(-3,-1)uu(1,2)` `(-2,-1)uu(2,oo)` `(-oo,-3)uu(-2,-1)uu(2,oo)` none of these

Options

A. `[-3,-1] cup [1,2]`
B. `(-2, -1) cup [2,oo)`
C. `(-oo,-3] cup (-2,-1) cup (2,oo)`
D. None of these

Answer

C

Explanation

Case I:
` 0 lt |x|-1 lt 1 " or " 1 lt |x| lt 2`
Then `x^(2)+4x+4 le 1`
or `x^(2)+4x+3 le 0`
or `-3 le x le -1`
So, `x in (-2,-1) " (1)" `
Case II:
` |x|-1 gt 1 " or " |x| gt 2`
Then `x^(2)+4x+4 ge 1`
or `x^(2)+4x+3 ge 0`
or `x ge -1 " or " x le -3`
So, ` x in (-oo, -3] cup (2,oo)`
So, ` x in (-oo, -3] cup (2,oo) " (2) " `
From (1) and (2), ` x in (-oo, -3] cup (-2, -1) cup (2,oo)`.