1. The domain of the function $f(x)=\frac{1}{\sqrt{-(\lg x)^{2}+3 \lg x-2}}$ is
$$
-(\lg x)^{2}+3 \lg x-2>0 \Rightarrow(\lg x)^{2}-3 \lg x+2<0 \Rightarrow 1<\lg x<2 \Rightarrow 10<x<100,
$$

Therefore, the domain of the function $f(x)$ is $(10,100)$.