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. $(d)$
Solution:
- Let $y=f(x)=\frac{x^{2}+x+2}{x^{2}+x+1}, x \in R$
$$ \begin{array}{ll} \therefore & y=\frac{x^{2}+x+2}{x^{2}+x+1} \\ & y=1+\frac{1}{x^{2}+x+1} \quad \text { [i.e. } y>1 \text { ] } \\ \Rightarrow & y x^{2}+y x+y=x^{2}+x+2 \\ \Rightarrow & x^{2}(y-1)+x(y-1)+(y-2)=0, \forall x \in R \\ \text { Since, } x \text { is real, } D \geq 0 \\ \Rightarrow & (y-1)^{2}-4(y-1)(y-2) \geq 0 \\ \Rightarrow & (y-1){(y-1)-4(y-2)} \geq 0 \end{array} $$
$$ \begin{array}{lr} \Rightarrow & (y-1)(-3 y+7) \geq 0 \\ \Rightarrow & 1 \leq y \leq \frac{7}{3} \end{array} $$
From Eqs. (i) and (ii), Range $\in 1, \frac{7}{3}$
