$\text{ Let }\mathrm{f}:(1,3)\to \mathrm{R}\text{ be a function defined by }\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}\left[\mathrm{x}\right]}{1+{\mathrm{x}}^2},\text{ where }\left[\mathrm{x}\right]\text{ denotes the greatest }$

integer$\le x$ . Then the range of f is:

$\begin{array}{l}f\left(x\right)=\left\{\begin{array}{lll}\frac{x}{x^2+1};& x\in (1,2)& \\ \frac{2x}{x^2+1};& x\in [2,3)& \end{array} \therefore f\left(x\right)\right.\text{ is a decreasing function }\\ \text{ If }x\in (1,2)\text{ then }f\left(x\right)\in \left(\frac{2}{5},\frac{1}{2}\right)\\ \text{ ff }x\in [2,3)\text{ then }f\left(x\right)\in \left(\frac{3}{5},\frac{4}{5}\right] \Rightarrow y\in \left(\frac{2}{5},\frac{1}{2}\right)\cup \left(\frac{3}{5},\frac{4}{5}\right]\end{array}$