Question

The natural domain of the function $f\left(x\right)=\sqrt{\left(sin\right)^{- 1} \left(2 x\right) + \frac{\pi }{3}}$ is

• A. $\left[- \frac{1}{2} , \frac{1}{2}\right]$
• B. $\left[- \frac{\sqrt{3}}{4} , \frac{1}{2}\right]$
• C. $\left[\frac{13}{4} , \frac{1}{2}\right]$
• D. $\left[- \frac{\sqrt{3}}{2} , 1\right]$

Answer 1

Answer: (B)

Here, $2x\in \left[- 1,1\right]$ and $\left(sin\right)^{- 1} \left(2 x\right)+\frac{\pi }{3}\geq 0$
Hence, $x\in \left[- \frac{1}{2} , \frac{1}{2}\right]$ and $x\geq -\frac{\sqrt{3}}{4}$
So the domain is $\left[- \frac{\sqrt{3}}{4} , \frac{1}{2}\right]$