Example 1. The function $y=\log _{0.5}\left(x^{2}+4 x+4\right)$ is increasing in what interval?
\[
\begin{array}{l}
\quad \text { Solve } y=\log _{0.5}\left(x^{2}+4 x+4\right) \\
=\log _{0.5}(x+2)^{2} \text { The domain is } x \neq-2 . \\
f_{2}(x)=(x+2)^{2} \text { is decreasing in }(-\infty,-2) , \\
f_{1}(x)=\log _{0.5} x \text { is decreasing in } f_{2}[(-\infty,-2)] \\
=(0,+\infty) \text { . Therefore, } y=\log _{0.5}\left(x^{2}+4 x+4\right) \text { is increasing in }(-\infty,-2) .
\end{array}
\]