Example 2 Find the interval of monotonic increase for the function $f(x)=\log _{\frac{1}{2}}\left(x^{2}-2 x-3\right)$.
Solve: From $x^{2}-2 x-3>0$, we get $x>3$ or $x<-1$. Let $y=f(u)=\log _{\frac{1}{2}} u$, $u=x^{2}-2 x-3$. Since $f(u)$ is a monotonically decreasing function on $(0,+\infty)$, and $u=x^{2}-2 x-3$ is a monotonically decreasing function on the interval $(-\infty,-1)$, it follows from the monotonicity of composite functions that the function $f(x)$ is monotonically increasing on the interval $(-\infty,-1)$. Similarly, the function $f(x)$ is monotonically decreasing on the interval $(3,+\infty)$. Therefore, the monotonic increasing interval of the function $f(x)=\log _{\frac{1}{2}}\left(x^{2}-2 x-3\right)$ is $(-\infty,-1)$.