Domain of definition of the real valued function \(f(x)=\sqrt{\sin ^{-1}(2 x)+\frac{\pi}{6}}\) is

\(\begin{aligned} & \\ & f(x)=\sqrt{\sin ^{-1}(2 x)+\frac{\pi}{6}} \text { to find domain } \\ & \\ & \sin ^{-1}(2 x)+\frac{\pi}{6} \geq 0 \\ & \\ & \text { But } \frac{-\pi}{2} \leq \sin ^{-1} \theta \leq \frac{\pi}{2} \\ & \\ & \frac{-\pi}{6} \leq \sin ^{-1}(2 x) \leq \frac{\pi}{2} \\ & \\ & \Rightarrow-\sin \frac{\pi}{6} \leq 2 x \leq \sin \frac{\pi}{2} \\ & \\ & \Rightarrow \frac{-1}{2} \leq 2 x \leq 1 \\ & \\ & \Rightarrow \frac{-1}{4} \leq x \leq \frac{1}{2} \\ & \therefore \quad \text { Domain of } \sqrt{\sin ^{-1}(2 x)+\frac{\pi}{6}} \text { is }\left[-\frac{1}{4}, \frac{1}{2}\right]\end{aligned}\)