89 If $f(x)=5x-3, x \in\{0,2,3,4,5\}, g(x)=x^{2}-2x-3, x \in\{-2$, $-1,1,4,5\}$, then the domain of the function $F(x)=\lg f(x)+\frac{1}{\sqrt{g(x)}}$ is $\qquad$
$89\{4,5\}$. According to the problem, we have
$$
\left\{\begin{array}{l}
f(x)=5 x-3>0 \\
g(x)=x^{2}-2 x-3>0
\end{array}\right.
$$

Solving this, we get $x>3$, so the domain of $F(x)$ is $\{4,5\}$.