9.112. For what values of $a$ is the inequality $\frac{a x}{x^{2}+4}<1$ true? If $x>0$, then we have $a x < x^{2} + 4$. This inequality holds for any values of $x \in R$ if $D=a^{2}-36<0$, $a^{2}<36$, $-6<a<6$.

Answer: $a \in(-6 ; 6)$.

9.113. Find the domain of the function $f$, if $f(x)=\sqrt{\log _{0.5}\left(x^{2}-9\right)+4}$.
## Solution.

The domain of the given function will be found by solving the inequality

$$
\begin{aligned}
& \log _{0.5}\left(x^{2}-9\right)+4 \geq 0, \log _{0.5}\left(x^{2}-9\right) \geq-4 \Leftrightarrow \\
& \Leftrightarrow 09\end{aligned} \Leftrightarrow\left\{\begin{array}{l}-5 \leq x \leq 5, \\ {\left[\begin{array}{l}x>3, \\ x<-3 .\end{array}\right.}\end{array}\right.\right.
$$

Answer: $x \in[-5 ;-3) \cup(3 ; 5]$.