$$\begin{aligned} & f(x)=\cos ^{-1}\left(\frac{4 x+5}{3 x-7}\right) \\ & \Rightarrow-1 \leq\left(\frac{4 x+5}{3 x-7}\right) \leq 1 \\ & \left(\frac{4 x+5}{3 x-7}\right) \geq-1 \\ & \frac{4 x+5+3 x-7}{3 x-7} \geq 0 \\ & \Rightarrow \frac{7 x-2}{3 x-7} \geq 0 \end{aligned}$$

$$\begin{aligned} & x \in\left(-\infty, \frac{2}{7}\right] \cup\left(\frac{7}{3}, \infty\right) \\ & \& \frac{4 x+5}{3 x-7} \leq 1 \Rightarrow \frac{x+12}{3 x-7} \leq 0 \end{aligned}$$

$\therefore$ Domain of $\mathrm{f}(\mathrm{x})$ is

$$\left[-12, \frac{2}{7}\right] \alpha=-12, \beta=\frac{2}{7}$$

$$g(x)=\log _2\left(2-6 \log _{27}(2 x+5)\right)$$

Domain

$$2-6 \log _{27}(2 x+5)>0$$

$$\begin{array}{ll} \Rightarrow & 6 \log _{27}(2 \mathrm{x}+5)<2 \\ \Rightarrow & \log _{27}(2 \mathrm{x}+5)<\frac{1}{3} \\ \Rightarrow & 2 \mathrm{x}+5<3 \\ \Rightarrow & \mathrm{x}<-1 \end{array}$$

$\& 2 x+5>0 \Rightarrow x>-\frac{5}{2}$

Domain is $\mathrm{x} \in\left(-\frac{5}{2},-1\right)$

$$\begin{aligned} &\gamma=-\frac{5}{2}, \delta=-1\\ &\begin{aligned} & |7(\alpha+\beta)+4(\gamma+\delta)|=\left\lvert\, 7\left(\left.-12+\frac{2}{7}+4\left(-\frac{5}{2}-1\right) \right\rvert\,\right.\right. \\ & |-82-14|=96 \end{aligned} \end{aligned}$$