$\begin{array}{l}
y = \sin x - \cos x + 3\\
y = \sqrt 2 \sin \left( {x - \dfrac{\pi }{4}} \right) + 3\\
- 1 \le \sin \left( {x - \dfrac{\pi }{4}} \right) \le 1\\
\Rightarrow - \sqrt 2 \le \sqrt 2 \sin \left( {x - \dfrac{\pi }{4}} \right) \le \sqrt 2 \\
\Rightarrow 3 - \sqrt 2 \le y \le 3 + \sqrt 2 \\
\Rightarrow \left\{ \begin{array}{l}
\min y = 3 - \sqrt 2 \\
\max y = 3 + \sqrt 2
\end{array} \right.
\end{array}$
