JEE Mathematics 2024 Question with Solution

Given:

• $f(x) = \frac{2x + 3}{2x + 1}$ with domain $\mathbb{R} \setminus \left\{-\frac{1}{2}\right\}$
• $g(x) = \frac{|x| + 1}{2x + 5}$ with domain $\mathbb{R} \setminus \left\{-\frac{5}{2}\right\}$

Find: The domain of $f(g(x))$.

For the composite function $f(g(x))$ to be defined, two conditions are needed:

1. $g(x)$ itself must be defined.
2. The value of $g(x)$ must belong to the domain of $f$, so $g(x) \ne -\frac{1}{2}$.

From $g(x) = \frac{|x| + 1}{2x + 5}$, we first require

so that

Now check whether $g(x) = -\frac{1}{2}$ is possible:

Cross-multiplying,

so

which gives

Consider cases for $|x|$.

If $x \ge 0$, then $|x| = x$. So

But this contradicts $x \ge 0$. Hence no solution arises from this case.

If $x < 0$, then $|x| = -x$. So

which is impossible. Hence there is no solution in this case either.

Therefore, $g(x) = -\frac{1}{2}$ is never attained for any valid real $x$. So no extra restriction is added beyond the domain of $g$.

Hence the domain of $f \circ g$ is

Therefore, the correct option is A.