Which of the following number(s) lie(s) in the range of the function $y = \frac{x^{2} + 14 x + 9}{x^{2} + 2 x + 3}, x \in R$ ?

2

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\frac{7}{2}

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- 4

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- 7

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Solution

The correct options are
A $2$
B $- 4$
C $\frac{7}{2}$
$y x^{2} + 2 x y + 3 y = x^{2} + 14 x + 9.$
$∴ x^{2} (y - 1) + x (2 y - 14) + 3 y - 9 = 0$
We have $x \in R, \Rightarrow (2 y - 14)^{2} - 4 (y - 1) (3 y - 9) \geq 0.$
$∴ y^{2} - 14 y + 49 - (3 y^{2} - 9 y - 3 y + 9) \geq 0.$
$∴ - 2 y^{2} - 2 y + 40 \geq 0.$
$\Rightarrow y^{2} + y - 20 \leq 0.$
$\Rightarrow (y + 5) (y - 4) \leq 0.$
$\Rightarrow y \in [- 5, 4]$
Hence, options A, B, C are correct.

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