First slide

Theory of expressions

Question

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

Moderate

A

$(1, \infty)$
B

(1, 3/2)
C

(1, 7/3]
D

(1, 7/5]

Solution

Let $y = \frac{x^{2} + x + 2}{x^{2} + x + 1} = 1 + \frac{1}{x^{2} + x + 1}$
$\begin{aligned} = 1 + \frac{1}{(x + 1 / 2)^{2} + 3 / 4} > 1 \\ \Rightarrow \frac{1}{y - 1} = x^{2} + x + 1 = {(x + \frac{1}{2})}^{2} + \frac{3}{4} \geq \frac{3}{4} \end{aligned}$
Thus, $y > 1$ and $y - 1 \leq 4 / 3 \Rightarrow y \leq 7 / 3$
$∴ 1 < y \leq 7 / 3$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App