Question

The domain of the function $f \left(x\right)=\frac{1}{\sqrt{\left\{sin\,x\right\}+\left\{sin\left(\pi+x\right)\right\}}}$ where $\{\cdot \}$ denotes fractional part, is

• A. $[0$, $\pi]$
• B. $(2n + 1)\pi/2$, $n \in Z$
• C. $(0$, $\pi)$
• D. None of these

Answer 1

The Correct Option is D

$f \left(x\right)=\frac{1}{\sqrt{\left\{sin\,x\right\}+\left\{sin\left(\pi+x\right)\right\}}}$ $=\frac{1}{\sqrt{\left\{sin\,x\right\}+\left\{-sin\,x\right\}}}$ Now, $\left\{sin x\right\}+\left\{-sinx\right\} = \begin{cases} 0, & \text{if $sin\,x$ is integer} \\[2ex] 1, & \text{if $sin\,x$ is not integer} \end{cases}$ For$ f(x)$ to be defined, $\{sinx\} + \{-sinx\} \neq 0$ $\Rightarrow sinx \neq$ integer $\Rightarrow sinx \ne \pm1$, $0$ $\Rightarrow x \ne\frac{n\pi}{2}$ Hence, domain is $R-\left\{\frac{n\pi}{2}, n\in I\right\}$.