CAT 2025 Slot 2 QA Question 11: Let $f(x) = \dfrac{x}{2 x - 1}$ and $g(x) = \dfrac{x}{x - 1}$. Then, the domain of the function $h(x) = f(g(x)) + g(f(x))$ is all real numbers except

Question

Let $f(x) = \dfrac{x}{2 x - 1}$ and $g(x) = \dfrac{x}{x - 1}$. Then, the domain of the function $h(x) = f(g(x)) + g(f(x))$ is all real numbers except

Accepted Answer

The function $f(x)$ is undefined at $x = \dfrac{1}{2}$, and $g(x)$ is undefined at $x = 1$.

For $f(g(x))$, $g(x)$ must be defined and must not equal $\dfrac{1}{2}$.
$\dfrac{x}{x - 1} = \dfrac{1}{2}$
$\Rightarrow 2 x = x - 1$
$\Rightarrow x = -1$

For $g(f(x))$, $f(x)$ must be defined and must not e

Let f(x)=x2x−1f(x) = \dfrac{x}{2 x - 1}f(x)=2x−1x​ and g(x)=xx−1g(x) = \dfrac{x}{x - 1}g(x)=x−1x​. Then, the domain of the function h(x)=f(g(x))+g(f(x))h(x) = f(g(x)) + g(f(x))h(x)=f(g(x))+g(f(x)) is all real numbers except

Let $f(x) = \dfrac{x}{2 x - 1}$ and $g(x) = \dfrac{x}{x - 1}$ . Then, the domain of the function $h(x) = f(g(x)) + g(f(x))$ is all real numbers except