Inverse Trigonometric Functions Question 64
Question: The range of the function $ f(x)=si{n^{-1}}(log[x])+log(si{n^{-1}}[x]); $ (Where [.] denotes the greatest integer function) is
Options:
A) $ R $
B) $ [1,2) $
C) $ { \log \frac{\pi }{2} } $
D) $ {-sin1} $
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Answer:
Correct Answer: C
$ {{\sin }^{-1}}(\log [x]) $ is defined if $ -1\le \log [x]\le 1 $ and $ [x]>0 $
$ \Rightarrow \frac{1}{e}\le [x]\le e\Rightarrow [x]=1,2\Rightarrow x\in [1,3) $
Again, $ \log ({{\sin }^{-1}}[x]) $ is defined if $ {{\sin }^{-1}}[x]>0 $ and $ -1\le [x]\le 1 $
$ \Rightarrow [x]>0and-1\le [x]\le 1\Rightarrow 0<[x]\le 1 $
$ \Rightarrow x\in [1,2) $
$ \therefore $ Domain of $ f(x)=[1,2) $ For $ 1\le x<2,[x]=1 $
$ \therefore f(x)=si{n^{-1}}0+\log \frac{\pi }{2}=\log \frac{\pi }{2},\forall x\in [1,2) $
$ \therefore $ Range of $ f(x)={ \log \frac{\pi }{2} } $
BETA
