Question

The Range of the function
$f\left(x\right)={\log }_{10}\left\{{\sin }^{-1}\left(\sqrt{x-5}\right)+\frac{3\pi }{2}\right\}$ is

• A. $\left[\log \frac{3\pi }{2},\log 2\pi \right]$
• B. $\left[\log \frac{3\pi }{2},\log 3\pi \right]$
• C. $\left[\log \frac{3\pi }{2},\log \pi \right]$
• D. $\left[\log \frac{3\pi }{4},\log 2\pi \right]$

Answer 1

The correct option is A $\left[\log \frac{3\pi }{2},\log 2\pi \right]$

$0\le \sqrt{x-5}\le 1\therefore x\in [5,6]{\sin }^{-1}\sqrt{x-5}\in [0,\frac{\pi }{2}]$

$\therefore$ Range of $f\left(x\right)\in [{\log }_{10}\frac{3\pi }{2},{\log }_{10}2\pi ]$