11.1. Find the domain and range of the function $\sqrt{1-\cos 2 x+2 \sin x}+\frac{1}{\sqrt{\sin ^{2} x+\sin x}}$.
Answer: The domain of definition $2 k \pi < x < \pi + 2 k \pi \Leftrightarrow \sin x(\sin x+1)>0 \Leftrightarrow$ $\sin x>0$ (since $\sin x \geq-1) \Leftrightarrow 2 k \pi < x < \pi + 2 k \pi (k \in \mathbb{Z})$. Further, investigating the function $y(t)$, we obtain the minimum point $t_{0}=2^{-1 / 4}<\sqrt{2}$ and the corresponding value $y\left(t_{0}\right)=2 \sqrt[4]{2}$ (see a similar solution in problem 11.2 from 19.12.2015). Hence, the result follows.