If the domain of the function f (x) = sin^{-1} (x - 1 / 2x + 3) is R - (α, β) then 12αβ is equal to : (1) 36 (2) 24

Question

If the domain of the function \(f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)\ \text{is}\ R-(\alpha, \beta)\) then \(12 \alpha \beta\) is equal to :

(1) 36

(2) 24

(3) 40

(4) 32

Answer 1

Correct option is : (4) 32 

Domain of \(f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)\)is

\(2 x+3 \neq 0 \ \&\ x \neq \frac{-3}{2}\)and \(\left|\frac{(x-1)}{2 x+3}\right| \leq 1\)

\(|x-1| \leq|2 x+3|\) 

For \(|2 x+3| \geq|x-1|\)

\(x \in(-\infty,-4] \cup\left(-\frac{2}{3}, \infty\right)\)

\(\alpha=-4 \ \&\ \beta=-\frac{2}{3}: 12 \alpha \beta=32\)