Question

Let $f\left(x\right)=\left[x\right]\sin \left(\frac{\pi }{\left[x+1\right]}\right)$, where $\left[x\right]$ denotes the greatest integer function. The domain of $f$ is

• A. $(-\infty ,-1)$
• B. $[0,\infty )$
• C. $(-\infty ,-1)\cup [0,\infty )$
• D. None of these

Answer 1

The correct option is C

$(-\infty ,-1)\cup [0,\infty )$

Explanation for the correct option.

Finding the domain of $f$

$f$ is defined only if the denominator of $\sin$term is not equal to zero

$\left[x+1\right]=0$ when

$$\begin{gathered} \Rightarrow 0\le (x+1)<1 \\ \Rightarrow -1\le x<0 \\ \Rightarrow R-[-1,0) \end{gathered}$$

Therefore, the domain of $f$ is $x\in (-\infty ,-1)\cup [0,\infty ).$

Therefore, the correct answer is option (C).