Question

Find the domain of $f\left(x\right)=\sin \left|x\right|+{\sin }^{-1}\left(\tan x\right)+\sin \left({\sin }^{-1}x\right)$

• A. $x\in [-\frac{\pi }{4},\frac{\pi }{4}]$
• B. $x\in [-\frac{\pi }{2},\frac{\pi }{2}]$
• C. $x\in [-\pi ,\pi ]$
• D. $x\in [0,\frac{\pi }{4}]$

Answer 1

The correct option is A $x\in [-\frac{\pi }{4},\frac{\pi }{4}]$

$f\left(x\right)=\sin \left|x\right|+{\sin }^{-1}\left(\tan x\right)+\sin \left({\sin }^{-1}x\right)$
For $\sin \left|x\right|$ $x\in R$ ...(1)
For ${\sin }^{-1}\left(\tan x\right)1\le \tan x\le 1\Rightarrow \frac{-\pi }{4}\le x\le \frac{\pi }{4}$ ...(2)
For $\sin \left({\sin }^{-1}x\right)-1\le x\le 1$ ...(3)

From (1), (2) and (3)
Domain is $[-\frac{\pi }{4},\frac{\pi }{4}]$