Question

The domain of the function $\sqrt{\left\{\frac{{\log }_{0.3}|x-2|}{\left|x\right|}\right\}}$ is

• A. $x\in [(1,2)\cup (2,3)]$
• B. $x\in [(1,3)\cup (3,5)]$
• C. $x\in [(0,2)\cup (2,3)]$
• D. $x\in (1,3)$

Answer 1

The correct option is A $x\in [(1,2)\cup (2,3)]$

The given function is defined if $x\ne 0,x\ne 2$ and
${\log }_{0.3}|x-2|>0$

Now ${\log }_{0.3}|x-2|>0$

$\Rightarrow |x-2|<1$ , as the base $0.3$ is less than $1$, the inequality is reversed.

$\therefore -1\le x-2\le 1$

$1\le x\le 3.$

Out of the above, we have to exclude $0$ and $2$.

Hence the domain is $1\le x\le 3$ but
$x\ne 0,x\ne 2$ or $x\in [(1,2)\cup (2,3)]$