Question

The domain of the function
$f\left(x\right)=\frac{1}{\sqrt{\left\{sinx\right\}+\left\{sin\left(\pi +x\right)\right\}}}$
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

Answer: (D)

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