Find the domain of `f(x) = sin^-1 [ log_9(x^2/4)]`

Explanation

We have
`f(x) ="sin"^(-1){log_(9)((x^(2))/(4))}`
Since the domain of ` sin^(-1)x` is `[-1,1]`. Therefore,
`f(x) ="sin"^(-1){log_(9)((x^(2))/(4))}`
is defined if
`-1 le log_(9)((x^(2))/(4)) le 1`
or ` 9^(-1) le (x^(2))/(4) le 9^(1)`
or `(4)/(9) le x^(2) le 36`
or `(2)/(3) le |x| le 6`
or `x in [-6, -(2)/(3)] cup [(2)/(3),6] " " ( :' a le |x| le b hArr x in [-b,-a] cup [a,b])`
Hence, the domain of f(x) is `[-6, -(2)/(3)] cup [(2)/(3),6]`.