Question

The range of the function $f(x)=\frac{x^{2}+x+2}{x^{2}+x+1}, x \in \mathbb{R}$ is:

A $(1, \infty)$ B $\left(1, \frac{11}{7}\right)$ C) $\left(1, \frac{7}{3}\right]$ D $\left(1, \frac{7}{5}\right)$

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Answer

:

The correct option is C $\left(1, \frac{7}{3}\right]$.

Let $ y = \frac{x^{2} + x + 2}{x^{2} + x + 1} = 1 + \frac{1}{x^{2} + x + 1} $.

Note that $ x^{2} + x + 1 = \left(x + \frac{1}{2}\right)^{2} + \frac{3}{4} $.

Since $ x \in \mathbb{R} $,

$$ 0 \leq \left(x + \frac{1}{2}\right)^{2} < \infty $$

Thus,

$$ \frac{3}{4} \leq x^{2} + x + 1 < \infty $$

Inverting this inequality, we get:

$$ 0 < \frac{1}{x^{2} + x + 1} \leq \frac{4}{3} $$

Therefore,

$$ 1 < 1 + \frac{1}{x^{2} + x + 1} \leq \frac{7}{3} $$

Hence,

$$ 1 < y \leq \frac{7}{3} $$

Thus, the range of the function $ f $ is $\left(1, \frac{7}{3}\right]$.


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