Mathematics Question from JEE Main 2023 · 12 April, Shift 1

Let $D$ be the domain of the function $f(x) = \sin^{-1} \left( \log_{3x} \left( \frac{6 + 2 \log_3 x}{-5x} \right) \right)$. If the range of the function $g: D \to \mathbb{R}$ defined by $g(x) = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer function, is $(\alpha, \beta)$, then $\alpha^2 + \frac{5}{\beta}$ is equal to: