Range of the function f ( x ) = x 2 + x + 2 x 2 + x + 1 , x R is [2003]

Question

Range of the function $f (x) = \frac{x^{2} + x + 2}{x^{2} + x + 1}, x \in R$ is [2003]

Smart Achievers · Accepted Answer

(3)

$f (x) = \frac{x^{2} + x + 2}{x^{2} + x + 1} = \frac{(x^{2} + x + 1) + 1}{x^{2} + x + 1}$

$= 1 + \frac{1}{{(x + \frac{1}{2})}^{2} + \frac{3}{4}}$

We can see here that as $x \to \infty, f (x) \to 1$ which is the minimum value of $f (x), i . e . f_{\min} = 1$ .

Also $f (x)$ is maximum when ${(x + \frac{1}{2})}^{2} + \frac{3}{4}$ is minimum, which is so when $x = - \frac{1}{2}$ .

$∴$ $f_{\max} = 1 + \frac{1}{\frac{3}{4}} = \frac{7}{3},$ $∴$ $R_{f} = (1, \frac{7}{3}]$

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