Question

The range of the function $f(x)\frac{x^2-x+1}{x^2+x+1}$ where $x\in R,$ is

• A. $(-\infty ,3]$
• B. $(-\infty ,\infty )$
• C. $[3,\infty )$
• D. $\left[\frac{1}{3},3\right]$
• E. $\left(-\infty ,\frac{1}{3}\right)\cup (3,\infty )$

Answer 1

Answer: (D)

Let $y=\frac{x^2-x+1}{x^2+x+1}$
$\Rightarrow$ $x^2(y-1)+4(y+1)+(y-1)=0$
Now, $D\ge 0$
$\Rightarrow$ ${(y+1)}^2-4{(y-1)}^2\ge 0$
$\Rightarrow$ $-3y^2-3+10y\ge 0$
$\Rightarrow$ $3y^2-10y+3\le 0$
$\Rightarrow$ $y=\frac{10\pm \sqrt{64}}{6}\le 0$ $\left(y-\frac{1}{3}\right)(y-3)\le 0$
$\Rightarrow$ $y\in \left[\frac{1}{3},3\right]$