Sets Relations And Functions Question 104
Question: The domain of the function $f(x)=\frac{1}{\sqrt{\{\sin x\}+\{\sin (\pi+x)\}}}$, where $ \{ \} $ denotes the fractional part, is
Options:
A) $ [0,\pi ] $
B) $ (2n+1)\pi /2,n\in Z $
C) $ (0,\pi ) $
D) none of these
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Answer:
Correct Answer: D
Solution:
- [d] $ f(x)=\frac{1}{\sqrt{{sinx}+{sin(\pi +x)}}} $
$ =\frac{1}{\sqrt{{sinx}+{-sinx}}} $
Now, $ {\sin x}+{-\sin x}= \begin{cases} 0, \text{if } \sin x \text{ is an integer } \\ 1,\text{if } \sin x \text{ is not an integer} \\ \end{cases} . $
For $ f(x) $ to get defined, $ {sinx}+{-sinx}\ne 0 $
Or $ \sin x\ne integer $
Or $ \sin x\ne \pm 1 $
Or $ x\ne \frac{n\pi }{2},n\in I $
Hence, the domain is $ \mathbb{R}-{ \frac{n\pi }{2},|,n\in \mathbb{Z} } $ .
BETA
