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) $(1,11 / 7)$
(c) $(1,7 / 3]$
(d) $(1,7 / 5)$
$(2003,2 M)$
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Answer:
Correct Answer: 4. (c)
Solution:
- Let $y=f(x)=\frac{x^{2}+x+2}{x^{2}+x+1}, x \in R$
$\therefore y=\frac{x^{2}+x+2}{x^{2}+x+1} $
$ y=1+\frac{1}{x^{2}+x+1} \quad $ [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 $
$\text { Since, } x \text { 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 $\in 1, \frac{7}{3}$
