JEE MAIN - Mathematics (2025 - 23rd January Morning Shift - No. 4)

Let $f(x)=\log _{\mathrm{e}} x$ and $g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}$. Then the domain of $f \circ g$ is

$(0, \infty)$

$[1, \infty)$

$\mathbb{R}$

$[0, \infty)$

$$\begin{aligned} & f(x)=\ln x \\ & g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1} \\ & D_g \in R \\ & D_f \in(0, \infty) \end{aligned}$$

For $D_{f o g} \Rightarrow g(x)>0$

$$\begin{aligned} & \frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}>0 \\ & \Rightarrow x^4-2 x^3+3 x^2-2 x+2>0 \end{aligned}$$

Clearly $\mathrm{x}<0$ satisfies which are included in option (1) only.