Official solution
\[f\left( x \right) = {\sin ^{ - 1}}\left( {{{\log }_3}\left( {{x \over 3}} \right)} \right)\] exists
if \[\,\,\,\, - 1 \le {\log _3}\left( {{x \over 3}} \right) \le 1\]
\[ \Leftrightarrow {3^{ - 1}} \le {x \over 3} \le {3^1}\]
\[ \Leftrightarrow 1 \le x \le 9\]
or \[\,\,\,\,x \in \left[ {1,9} \right]\]