Điều kiện xác định: $\cos x\ge -1\Rightarrow x\in \mathbb{R}$
$\begin{array}{l}
- 1 \le \cos x \le 1\\
\Rightarrow 0 \le \cos x + 1 \le 2\\
\Rightarrow 0 \le \sqrt {\cos x + 1} \le \sqrt 2 \\
\Rightarrow 0 \le 2\sqrt {\cos x + 1} \le 2\sqrt 2 \\
\Rightarrow - 3 \le y \le 2\sqrt 2 - 3\\
\Rightarrow \left\{ \begin{array}{l}
\min y = - 3 \Rightarrow \cos x = - 1\\
\max y = 2\sqrt 2 - 3 \Rightarrow \cos x = 1
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
\min y = - 3 \Rightarrow x = \pi + k2\pi \\
\max y = 2\sqrt 2 - 3 \Rightarrow x = k2\pi
\end{array} \right.\left( {k \in Z} \right)
\end{array}$
