The domain of definition of the function $y(x)$ given by $2^x + 2^y = 2$ is

If the domain of the function $f(x) = \log_e\left(\frac{2x-3}{5+4x}\right) + \sin^{-1}\left(\frac{4+3x}{2-x}\right)$ is $[\alpha, \beta)$,then $\alpha^2 + 4\beta$ is equal to

If $f(x) = 3 - x^2$ for $1 \le x \le 4$,then the domain of $\log_e(f(2x))$ is:

The domain of the function $f(x) = \frac{1}{\sqrt{|x|-x}}$ is

If the domain of the function $f(x) = \cos^{-1}\left(\frac{2-|x|}{4}\right) + (\log_e(3-x))^{-1}$ is $[-\alpha, \beta) \setminus \{\gamma\}$,then $\alpha + \beta + \gamma$ is equal to:

Domain of the function $\log |x^2 - 9|$ is