Question

The domain of the function $f\left(x\right)={\left({\log }_3\frac{1}{|\cos x|}\right)}^{\frac{1}{2020}}$ is

• A. $R-\{\frac{n\pi }{2},n\in Z\}$
• B. $R-\{n\pi ,n\in Z\}$
• C. $R$
• D. $R-\left\{\right(2n+1)\frac{\pi }{2},n\in Z\}$

Answer 1

The correct option is D $R-\left\{\right(2n+1)\frac{\pi }{2},n\in Z\}$

For $f\left(x\right)$ to be defined,
${\log }_3\frac{1}{\left(|\cos x|\right)}\ge 0(\because$we have even exponential root here)

$\Rightarrow \frac{1}{|\cos x|}\ge 3^0$

$\Rightarrow \frac{1}{|\cos x|}\ge 1$

$\Rightarrow |\cos x|\le 1$
and, $|\cos x|\ne 0\Rightarrow x\ne (2n+1)\frac{\pi }{2},n\in Z$.
$\therefore x\in R-\left\{\right(2n+1)\frac{\pi }{2},n\in Z\}$