$f :\left(1,3\right)\to R, f\left(x\right)=\frac{x\left[x\right]}{1+x^2}$

$f\left(x\right)=\left\{\begin{array}{ll}\frac{\mathrm{x}}{1+x^2},& x\in \left(1, 2\right)\\ \frac{2x}{1+x^2},& x\in [2, 3)\end{array}\right.$

$f^{\text{'}}\left(x\right)=\left\{\begin{array}{ll}\frac{\left(1+x^2\right)\left(1\right)-x\left(2x\right)}{{\left(1+x^2\right)}^2},& x\in \left(1, 2\right)\\ \frac{\left(1+x^2\right)\left(2\right)-2x\left(2x\right)}{{\left(1+x^2\right)}^2},& x\in [2,3)\end{array}\right.$

$f^{\text{'}}\left(x\right)=\left\{\begin{array}{ll}\frac{1-x^2}{1+x^2},& x\in \left(1, 2\right)\\ \frac{1-2x^2}{1+x^2},& x\in [2,3)\end{array}\right.$

$\therefore f\left(x\right)$ is decreasing function.
$\therefore {\mathrm{R}}_{\mathrm{f}} \in \left(\frac{2}{5}, \frac{1}{2}\right)\cup \left(\frac{6}{10}, \frac{4}{5}\right]$