Question ID: #228
Let $f(x)=\log_{e}x$ and $g(x)=\frac{x^{4}-2x^{3}+3x^{2}-2x+2}{2x^{2}-2x+1}$. Then the domain of $fog$ is
- (1) $\mathbb{R}$
- (2) $(0,0)$
- (3) $[0, \infty)$
- (4) $[1, \infty)$
Solution:
Domain of $f(g(x))$ requires $g(x) > 0$.
$$g(x) = \frac{x^4-2x^3+3x^2-2x+2}{2x^2-2x+1}$$
Numerator: $N(x) = (x^2-x)^2 + 2(x^2-x) + 2 = ((x^2-x)+1)^2 + 1 > 0 \quad \forall x \in \mathbb{R}$
Denominator: $D(x) = 2x^2-2x+1$.
Discriminant $\Delta = 4 – 8 = -4 < 0$ and $a=2>0$,
so $D(x) > 0 \quad \forall x \in \mathbb{R}$.
Since $N(x) > 0$ and $D(x) > 0$, $g(x) > 0$ for all $x \in \mathbb{R}$.
Thus, the domain is $\mathbb{R}$.
Ans. (1)
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