Mathematics - Functions Question with Solution | TestHub

$\begin{array}{rl}& \text{Let }y=\frac{x^2-x+1}{x^2+x+1}\\ & \Rightarrow {\mathrm{x}}^2(\mathrm{y}-1)+\mathrm{x}(\mathrm{y}+1)+(\mathrm{y}-1)=0\\ & \Rightarrow x=\frac{-(y+1)\pm \sqrt{(y+1{)}^2-4(y-1{)}^2}}{2(y-1)}\\ & =\frac{-(y+1)\pm \sqrt{-3y^2+10y-3}}{2(y-1)}\text{ is real iff }\\ & \mathrm{y}-1\ne 0\Rightarrow \mathrm{y}\ne 1\\ & \text{ If }y=1\text{ then original equation gives }\mathrm{x}=0\text{, }\\ & \text{ so taking }\mathrm{y}=1\\ & \text{ Also }3y^2-10y+3\le 0\\ & \Rightarrow (3y-1)(y-3)\le 0\\ & \Rightarrow \mathrm{y}\in \left[\frac{1}{3},3\right]\therefore \text{ Range is }\left[\frac{1}{3},3\right]\\ & \end{array}$