Example 3 Identify the monotonicity of the function $y=\sqrt{x^{2}+2 x-3}$.

Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. 

Example 3 Identify the monotonicity of the function $y=\sqrt{x^{2}+2 x-3}$.
Solve: From $x^{2}+2 x-3 \geqslant 0$ we get that the domain of the function is $\{x \mid x \leqslant-3$ or $x \geqslant 1\}$, and $y=$ $\sqrt{(x+1)^{2}-4}$, so the function $y=\sqrt{x^{2}+2 x-3}$ is monotonically decreasing on $(-\infty,-3]$ and monotonically increasing on $[1, +\infty)$.

Key Points for Solving: All properties of a function are based on its domain, therefore, to determine the monotonicity of the function, it is necessary to first determine the domain of the function.