Find the domain of $f(x) = \sin^{-1} {\log_9(x^2/4)}$

$[-6,-\frac{2}{3}]∪[\frac{2}{3},6]$ 

We have 

$f(x)=\sin^{-1}\{\log_9(\frac{x^2}{4})\}$

Since the domain of $\sin^{-1} x$ is [-1, 1]. Therefore,

$f(x)=\sin^{-1}\{\log_9(\frac{x^2}{4})\}$ is defined if

$-1≤\log_9(\frac{x^2}{4})≤1$

or $9^{-1}≤\frac{x^2}{4}≤9^1$

or $\frac{4}{9}≤x^2≤36$

or $\frac{2}{3}≤|x|≤6$

or $x∈[-6,-\frac{2}{3}]∪[\frac{2}{3},6]$  $(∵a≤|x|≤b⇔x∈ [-b, -a]∪[a, b])$

Hence, the domain of f(x) is $[-6,-\frac{2}{3}]∪[\frac{2}{3},6]$