SATHEE: Functions 1 Question 4

Functions 1 Question 4

4.

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

(a) $(1, \infty)$

(b) $(1, 11 / 7)$

(d) $(1, 7 / 5)$

$(2003, 2 M)$

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Answer:

Correct Answer: 4. (c)

Solution:

Let $y = f (x) = \frac{x^{2} + x + 2}{x^{2} + x + 1}, x \in R$

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

$y = 1 + \frac{1}{x^{2} + x + 1}$ [i.e. $y > 1$ ]

$\Rightarrow y x^{2} + y x + y = x^{2} + x + 2$

$\Rightarrow x^{2} (y - 1) + x (y - 1) + (y - 2) = 0, \forall x \in R$

$Since, x is real, D \geq 0$

$\Rightarrow (y - 1)^{2} - 4 (y - 1) (y - 2) \geq 0$

$\Rightarrow (y - 1) (y - 1) - 4 (y - 2) \geq 0$

$\Rightarrow (y - 1) (- 3 y + 7) \geq 0$

$\Rightarrow 1 \leq y \leq \frac{7}{3}$

From Eqs. (i) and (ii), Range $\in 1, \frac{7}{3}$

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Functions 1 Question 4

4.

Answer:

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