Let and . Then the domain of fog is [2025]

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Solution

Q.
Let $f (x) = \log_{e} x$ 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 fog is [2025]

1 $(0, \infty)$

2 $[0, \infty)$

3 $[1, \infty)$

4 R

Ans.
(4)

Given, $g (x) = \frac{x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 2}{2 x^{2} - 2 x + 1}$

Here, $D g \in R$ [ $∵ 2 x^{2} - 2 x + 1 > 0$ ]

Also, $f (x) = \log_{e} x \Rightarrow D_{f} \in (0, \infty)$ $∴ D_{f o g} \Rightarrow g (x) > 0$

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

The above expression is always positive for any value of x.

$∴ Domain of f o g \in R$ .

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