Question

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

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

Answer 1

Answer: (C)

$sin^{-1}\left(log\left[x\right]\right)$ is defined if $-1\le log\left[x\right]\le1 and \left[x\right]>0$
$\Rightarrow \frac{1}{e}\le\left[x\right]\le e \Rightarrow \left[x\right]=1, 2 \Rightarrow x\in[1,3)$
Again, $log\left(sin^{-1}\left[x\right]\right)$ is defined if
$sin^{-1}\left[x\right]>0 and -1\le\left[x\right]\le1$
$\Rightarrow [x] > 0 \,\, and \,\,-1 \le [x] \le \,\, \Rightarrow 0 < [x] \le 1$
$\Rightarrow x\in[1, 2)$
$\therefore $ Domain of $f \left(x\right) = [1,2)$
For $1\le x<2, \left[x\right]=1
\therefore f \left(x\right)=sin^{-1}\,0+log \frac{\pi}{2}=log \frac{\pi}{2}, \forall x\in[1,2)$
$\therefore $ Range of $\left(x\right)=\left\{log \frac{\pi}{2}\right\}$