QJEEM21Ma0262: JEE Main 2021 Mathematics - Class 12 JEEM21Ma0262 Solution: Let and . If , then the domain of the function fog is :

Question

Verified Solution • 2025–26 Session moderate

Let

f(x) = {\sin ^{ - 1}}x

and

g(x) = {{{x^2} - x - 2} \over {2{x^2} - x - 6}}

. If

g(2) = \mathop {\lim }\limits_{x \to 2} g(x)

, then the domain of the function fog is :

Accepted Answer

g(2) = \mathop {\lim }\limits_{x \to 2} {{(x - 2)(x + 1)} \over {(2x + 3)(x - 2)}} = {3 \over 7}

Domain of

fog(x) = {\sin ^{ - 1}}(g(x))

\Rightarrow |g(x)|\, \le 1

\left| {{{{x^2} - x - 2} \over {2{x^2} - x - 6}}} \right| \le 1

\left| {{{(x + 1)(x - 2)} \over {(2x + 3)(x - 2)}}} \right| \le 1

{{x + 1} \over {2x + 3}} \le 1

and

{{x + 1} \over {2x + 3}} \ge - 1

{{x + 1 - 2x - 3} \over {2x + 3}} \le 0

and

{{x + 1 + 2x + 3} \over {2x + 3}} \ge 0

{{x + 2} \over {2x + 3}} \ge 0

and

{{3x + 4} \over {2x + 3}} \ge 0

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

Step-by-Step Verified Solution