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