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

Answer: (D)

None of these

Answer 2

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,xis integer} \\\[2ex] 1, & \text{ifsin,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\\}.