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  4. If x is real, then the range of (x2 + 2x + 1/x2 + 2x + 7) is

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Q. If $x$ is real, then the range of $\frac{x^2 + 2x + 1}{x^2 + 2x + 7}$ is

AP EAMCETAP EAMCET 2018

Solution:

Let $\frac{x^{2}+2 x+1}{x^{2}+2 x+7}=y$,
$\because y \neq 1$ ...(i)
$\Rightarrow (y-1) x^{2}+2(y-1) x+(7 y-1)=0$
$\because x \in R$
so, $D \geq 0$
$\Rightarrow 4(y-1)^{2}-4(y-1)(7 y-1) \geq 0$
$\rightarrow (y-1)[y-1-7 y+1] \geq 0$
$\Rightarrow y(y-1) \leq 0$ ...(ii)
$\Rightarrow y \in[0,1]$
From Eqs. (i) and (ii), we are getting
$y \in[0,1)$