Relations Question 19
Question: The domain of the functions $ f(x)=\frac{1}{\sqrt{{ sinx }+{ \sin (\pi +x) }}} $ where $ { \cdot } $ denotes the fractional part, is
Options:
A) $ [0,\pi ] $
B) $ (2n+1)\pi /2,n\in Z $
C) $ (0,\pi ) $
D) none of these
Show Answer
Answer:
Correct Answer: D
Solution:
$ f(x)=\frac{1}{\sqrt{{sinx}+{sin(\pi +x)}}} $
$ =\frac{1}{\sqrt{{sinx}+{-sinx}}} $ Now, $ {\sin x}+{-\sin x}= \begin{cases} 0,\sin x\text{ is an integer} \ 1,\sin x\text{ is not an integer} \ \end{cases} . $
For $ f(x) $ to be defined, $ {sinx}+{-sin\ x}\ne 0 $ Or $ \sin x\ne integer $ Or $ \sin x\ne \pm 1,0 $ Or $ x\ne \frac{n\pi }{2},n\in \mathbb{Z} $
Hence, the domain is $ R-{ \frac{n\pi }{2}/n\in I } $ .
BETA
