Question

The domain of the function $f\left(x\right)=\frac{1}{\sqrt{|[|x|-5]|-5}}$ is
(where [.] denotes the greatest integer function)

• A. $(-\infty ,-5]$
• B. $(-\infty ,-11]\cup [11,\infty )$
• C. $(-\infty ,-5]\cup [5,\infty )$
• D. $[11,\infty )$

Answer 1

The correct option is B $(-\infty ,-11]\cup [11,\infty )$

For function to be defined
$|[|x|-5]|-5>0\Rightarrow |\left[|x|\right]-5|-5>0$
Now,
$|\left[|x|\right]-5|-5>0\Rightarrow |\left[|x|\right]-5|>5\Rightarrow \left[|x|\right]-5>5\text{or}\left[|x|\right]-5<-5\Rightarrow \left[|x|\right]>10\text{or}\left[|x|\right]<0$

$|x|\ge 11\Rightarrow x\in (-\infty ,-11]\cup [11,\infty )$

$\left[|x|\right]<0$
No solution possible

$\therefore x\in (-\infty ,-11]\cup [11,\infty )$