Question

The natural domain of the function $f\left(x\right)=\sqrt{{\left(sin\right)}^{-1}\left(2x\right)+\frac{\pi }{3}}$ 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 ${\left(sin\right)}^{-1}\left(2x\right)+\frac{\pi }{3}\ge 0$
Hence, $x\in \left[-\frac{1}{2},\frac{1}{2}\right]$ and $x\ge -\frac{\sqrt{3}}{4}$
So the domain is $\left[-\frac{\sqrt{3}}{4},\frac{1}{2}\right]$