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. $\left[ \frac{1}{3},3 \right]$

Answer 1

Answer: (D)

$\left[ \frac{1}{3},3 \right]$

Answer 2

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