[ If the domain of the function f(x)= _e((2 x-3)/(5+4 x))+ ^-1((4+3 x)/(2-x)) is [, ) , th…

Question

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

Accepted Answer

$\frac{2 x-3}{4 x+5}>0$

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

$\begin{aligned} & \Rightarrow \quad \frac{3 x+4-2+x}{x-2} \geq 0 \ & \Rightarrow \quad \frac{4 x+2}{x-2} \geq 0 \ & \Rightarrow \quad x \in\left(-\infty,-\frac{1}{2}\right] \cup(2, \infty)........(ii)\end{aligned}$

$$ \begin{aligned} & \frac{3 x+4}{2-x} \geq-1 \ \Rightarrow & \frac{3 x+4}{2-x}+1 \geq 0 \ \Rightarrow & \frac{3 x+4+2-x}{2-x} \geq 0 \ \Rightarrow & \frac{2 x+6}{x-2} \leq 0 \ \therefore & x \in[-3,2)........(iii) \end{aligned} $$

Taking intersection of (i), (ii) and (iii)

$$ \begin{aligned} & x \in\left[-3,-\frac{5}{4}\right) \ & \alpha=-3, \beta=-\frac{5}{4} \ & \alpha^2+4 \beta=4 \end{aligned} $$

Solutions · 1