Question

$f\left(x\right)=\frac{\sqrt{\left|\tan x\right|+\tan x}}{\surd 3x}$ is defined for

• A. $R$
• B. $R-\left\{\frac{1}{3}\right\}$
• C. $R-\left\{n\pi +\left(\frac{\pi }{2}\right):n\in I^{+}\right\}$
• D. None of these

Answer 1

The correct option is C

$R-\left\{n\pi +\left(\frac{\pi }{2}\right):n\in I^{+}\right\}$

Explanation for the correct option:

Find the domain of given function:

$\tan x$ is defined if $x\ne n\pi +\left(\frac{\mathrm{\pi }}{2}\right)$ — (1)

If $\tan x>0$, then $\left|\tan x\right|+\tan x>0$ — (2)

If $\tan x\le 0$, then $\left|\tan x\right|+\tan x=0$ — (3)

$\Rightarrow$Numerator is defined and non-zero for both (2) and (3)

$\therefore$Domain, ${\mathit{D}}_{\mathbf{f}}\mathbf{=}\mathit{R}\mathbf{–}\mathit{n}\mathit{\pi }\mathbf{+}\left(\frac{\pi }{2}\right)\mathbf{:}\mathit{n}\mathbf{\in }\mathit{I}\mathbf{+}$

Hence, Option ‘C’ is Correct.