Question

Find the interval the expression $\frac{x^2+2x+1}{x^2+2x+7}$ lies$\left(xϵR\right).$

• A. $[0,-1]$
• B. $[0,1)$
• C. $(0,1)$
• D. none of these

Answer 1

The correct option is B $[0,1)$

Let $y=\frac{x^2+2x+1}{x^2+2x+7}$

$(y-1)x^2+2(y-1)x+(7y-1)=0$

Given that $x\in R$

$\therefore 4(y-1{)}^2-4(y-1\left)\right(7y-1)\ge 0$

$(y-1)(y-1-7y+1)\ge 0$

$y(y-1)=0$

since $y\in [0,1]$

But $y\ne$( As coefficient of $x^2\ne 0$)

$\therefore y\in [0,1)$

$\therefore \frac{x^2+2x+1}{x^2+2x+7}\in [0,1)$