Question

$Rangeofthefunctionf\left(x\right)=\frac{x^2+x+2}{x^2+x+1};xϵRis$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$Wehave,f\left(x\right)=\frac{x^2+x+2}{x^2+x+1}=\frac{(x^2+x+1)}{x^2+x+1}=1+\frac{1}{{(x+\frac{1}{2})}^2+\frac{3}{4}}Wecanseeherethatasx\to \infty ,f\left(x\right)\to 1whichistheminvalueoff\left(x\right).Alsof\left(x\right)ismaxwhen{(x+\frac{1}{2})}^2+\frac{3}{4}isminwhichissowhenx=-\frac{1}{2}andthen\frac{3}{4}.\therefore f_{max}=1+\frac{1}{\frac{3}{4}}=\frac{7}{3}\therefore R_i=(1,\frac{7}{3}]$