Question

The domain of the function $f\left(x\right)=\sqrt{\frac{|x|-2}{|x|-5}}$ is

• A. $(-\infty ,-5)\cup (5,\infty )$
• B. $(-\infty ,-5)\cup [-2,2]\cup (5,\infty )$
• C. $[-2,2]$
• D. $(-\infty ,-5]\cup [-2,2]\cup [5,\infty )$

Answer 1

The correct option is B $(-\infty ,-5)\cup [-2,2]\cup (5,\infty )$

For $f\left(x\right)$ to be defined,
$\frac{|x|-2}{|x|-5}\ge 0\text{and}|x|-5\ne 0$
$\Rightarrow |x|-2\ge 0\text{and}|x|-5>0$
$\text{or}|x|-2\le 0\text{and}|x|-5<0$
$\Rightarrow x\in (-\infty ,-5)\cup (5,\infty )$
$\text{or}x\in [-2,2]$
$\therefore D_f=(-\infty ,-5)\cup [-2,2]\cup (5,\infty )$