Question

Let $f(x)=\sin ^{-1} x$ and $g(x)=\frac{x^{2}-x-2}{2 x^{2}-x-6} .$ If $g (2)=\displaystyle\lim _{ x \rightarrow 2} g ( x ),$ then the domain of the function $f og$ is :

• A. $(-\infty,-2] \cup\left[-\frac{3}{2}, \infty\right)$
• B. $(-\infty,-2] \cup[-1, \infty)$
• C. $(-\infty,-2] \cup\left[-\frac{4}{3}, \infty\right)$
• D. $(-\infty,-1] \cup[2, \infty)$

Answer 1

Answer: (C)

Domain of $\operatorname{fog}( x )=\sin ^{-1}( g ( x ))$
$\Rightarrow |g( x ) | \leq 1 \,\,\,\,\, g (2)=\frac{3}{7}$
$\left|\frac{ x ^{2}- x -2}{2 x ^{2}- x -6}\right| \leq 1$
$\left|\frac{( x +1)( x -2)}{(2 x +3)( x -2)}\right| \leq 1$
$\frac{ x +1}{2 x +3} \leq 1$ and $\frac{ x +1}{2 x +3} \geq-1$
$\frac{ x +1-2 x -3}{2 x +3} \leq 0$ and $\frac{ x +1+2 x +3}{2 x +3} \geq 0$
$\frac{ x +2}{2 x +3} \geq 0$ and $\frac{3 x +4}{2 x +3} \geq 0$
$x \in(-\infty,-2] \cup\left[-\frac{4}{3}, \infty\right)$