Question

The domain of definition of the function $f\left(x\right)=\frac{1}{\sqrt{\left|[\parallel x\parallel -1]\right|-5}}$ is

• A. $(-\infty ,-7]\cup [7,\infty )$
• B. $(-\infty ,\infty )$
• C. $(7,\infty )$
• D. $(-7,\infty )$

Answer 1

The correct option is A $(-\infty ,-7]\cup [7,\infty )$

$f\left(x\right)=\frac{1}{\sqrt{\left|\left[|x|-1|\right]\right|-5}}$

For the function to be defined

$\left|[|x|-1]\right|-5>0\Rightarrow \left|[|x|-1]\right|>5\Rightarrow -5>[|x|-1]>5\Rightarrow -5>|x|-1\ge 6$

$\Rightarrow -4>|x|\ge 7$

$-4>|x|$ is not possible

$\Rightarrow |x|\ge 7$

$\Rightarrow -7\ge x\ge 7$

$\Rightarrow x\in (-\infty ,-7]\cup [7,\infty )$

Hence, option $A$ is correct.