Question

Find the domain and range of $f\left(x\right)={\sin }^{-1}\left(\log \right[x\left]\right)+\log \left({\sin }^{-1}\right[x\left]\right),$ where $[\cdot ]$ denotes the greatest integer function

• A. Range of $f\left(x\right)=\in \left\{\log \frac{\pi }{2}\right\}$
• B. domain $\in [1,2)$
• C. domain $\in [-1,2)$
• D. Range of $f\left(x\right)=\in \left\{\log \frac{\pi }{4}\right\}$

Answer 1

Answer: (A), (B)

domain $\in [1,2)$
Range of $f\left(x\right)=\in \left\{\log \frac{\pi }{2}\right\}$

Given, $f\left(x\right)={\sin }^{-1}\left(\log \right[x\left]\right)+\log \left({\sin }^{-1}\right[x\left]\right)$

For given function to exist $0<\left[x\right]\le 1\Rightarrow 1\le x<2\Rightarrow \left[x\right]=1$
Thus domain is $(1,2)$
And range $={\sin }^{-1}0+\log \frac{\pi }{2}=\log \frac{\pi }{2}$