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…

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
  1. $[0, \infty)$
  2. $[1, \infty)$
  3. $(0, \infty)$
  4. $\mathbb{R}$

Solution

$f(g(x))=\ln \left(\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}\right)$
Since $2 x^2-2 x+1>0 \quad \forall x \in \mathbb{R} \because(-2)^2-4(2) < 0$
Consider
$\begin{aligned}
& x^4-2 x^3+3 x^2-2 x+2 \\ & =\left(x^4-2 x^3+x^2\right)+\left(x^2-2 x+1\right)+\left(1+x^2\right) \\ & =x^2(x-1)^2+(x-1)^2+\left(x^2+1\right)>0 \forall x \in \mathbb{R} \\ & \Rightarrow g(x)>0 \forall x \in \mathbb{R} \\ & \Rightarrow \ln f((x)), f(x)>0 \forall x \in \mathbb{R} \\ & \Rightarrow x \in \mathbb{R} \text { is domain }
\end{aligned}$

Asked in: JEE Main 2025 (23 Jan Shift 1)

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