Let f(x)=


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

  1. (1) $(-\infty,-2) \cup\left[-\frac{4}{3}, \infty\right)$

  2. (2) $(-\infty,-1] \cup[2, \infty)$

  3. (3) $(-\infty,-2] \cup[-1, \infty)$

  4. (4) $(-\infty,-2] \cup\left[-\frac{3}{2}, \infty\right)$

Correct Option: 1


$g(2)=\lim _{x \rightarrow 2} \frac{(x-2)(x+1)}{(2 x+3)(x-2)}=\frac{3}{7}$

For domain of fog $(\mathrm{x})\left|\frac{x^{2}-x-2}{2 x^{2}-x-6}\right| \leq 1 \Rightarrow(x+1)^{2} \leq(2 x+3)^{2} \Rightarrow 3 x^{2}+10 x+8 \geq 0$

$\Rightarrow(3 x+4)(x+2) \geq 0$

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

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