Question

The range of the function
$f(x)=\sin ^{-1}(\log [x])+\log \left(\sin ^{-1}[x]\right) ;$ (where [.] denotes the greatest integer function) is

• A. $R$
• B. $[1,2)$
• C. $\left\{\log \frac{\pi}{2}\right\}$
• D. $\{-\sin 1\}$

Answer 1

Answer: (C)

$\sin ^{-1}(\log [x])$ is defined if $-1 \leq \log [x] \leq 1$ and $[x]>0$
$\Rightarrow \frac{1}{e} \leq[x] \leq e $
$ \Rightarrow [x]=1,2 $
$ \Rightarrow x \in[1,3)$
Again, $\log \left(\sin ^{-1}[x]\right)$ is defined if
$\sin ^{-1}[x]>0$ and $-1 \leq[x] \leq 1$
$\Rightarrow [x]>0$ and $-1 \leq[x] \leq 1 $
$\Rightarrow 0<[x] \leq 1 $
$ \Rightarrow x \in[1,2)$
$\therefore $ Domain of $f(x)=[1,2)$
For $1 \leq x<2,[x]=1$
$\therefore f(x)=\sin ^{-1} 0+\log \frac{\pi}{2}=\log \frac{\pi}{2}, \forall x \in[1,2)$
$\therefore $ Range of $f(x)=\left\{\log \frac{\pi}{2}\right\}$