Question

Find the range of

$f\left(x\right)=\frac{x-1}{x^2-2x+3}$

Answer 1

Let $y=\frac{x-1}{x^2-2x+3}$
$\Rightarrow y{x}^2-(2y+1)x+(3y+1)=0$
Now since $x\in R$, so discriminant of above quadratic equation is $\ge 0$
$\Rightarrow (2y+1{)}^2-4y(3y+1)\ge 0$
$\Rightarrow 4y^2+4y+1-12y^2-4y\ge 0$
$\Rightarrow 1-8y^2\ge 0$

$\Rightarrow y^2\le \frac{1}{8}$
$\therefore y\in [-\frac{1}{2\sqrt{2}},\frac{1}{2\sqrt{2}}]$
which is range of $f\left(x\right)$