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

• A. $R$
• B. $[1,2)$
• C. $\left\{log\frac{\pi }{2}\right\}$
• D. $\left\{-sin1\right\}$

Answer 1

Answer: (C)

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