Let $$[\cdot]$$ denote the greatest integer function. If the domain of the function $$f(x) = \sin^{-1}\left(\frac{x + [x]}{3}\right)$$ is $$[\alpha, \beta)$$, then $$\alpha^2 + \beta^2$$ is equal to:

$$-1 \le \frac{x + \lfloor x \rfloor}{3} \le 1$$

$$\Rightarrow -3 \le x + \lfloor x \rfloor \le 3$$

$$\Rightarrow -3 - \lfloor x \rfloor \le x \le 3 - \lfloor x \rfloor$$

$$\text{If } \lfloor x \rfloor = -1 \Rightarrow x \in [-1, 0)$$

$$\text{If } \lfloor x \rfloor = 0 \Rightarrow x \in [0, 1)$$

$$\text{If } \lfloor x \rfloor = 1 \Rightarrow x \in [1, 2)$$

$$\text{Hence } x \in [-1, 2)$$

$$\Rightarrow \alpha = -1 \text{ and } \beta = 2$$

$$\Rightarrow \alpha^2 + \beta^2 = (-1)^2 + 2^2 = 5$$