Question

Find the range of $f\left(x\right)={\cos }^{-1}\sqrt{{\log }_{\left[x\right]}\frac{\left|x\right|}{x}},,$ where $[\cdot ]$denotes the greatest integer.

• A. $\left\{\frac{\pi }{2}\right\}$
• B. $\left\{\frac{\pi }{4}\right\}$
• C. $\left\{\frac{\pi }{6}\right\}$
• D. $\left\{\frac{\pi }{8}\right\}$

Answer 1

The correct option is A $\left\{\frac{\pi }{2}\right\}$

For given function to be defined,
$\frac{|x|}{x}>0\Rightarrow x>0$
Also $\left[x\right]>0,\ne 1\Rightarrow x\ge 2$
Thus $f\left(x\right)={\cos }^{-1}\sqrt{{\log }_{x\ge 2}1}={\cos }^{-1}0=\frac{\pi }{2}$
Hence range of f is $\left\{\frac{\pi }{2}\right\}$