JEE Main 2020 (Online) 8th January Evening ShiftQuestion 1
Let $f:(1,3)\to\mathbf{R}$ be a function defined by $f(x)=\frac{x[x]}{1+x^2}$, where $[x]$ denotes the greatest integer $\le x$. Then the range of $f$ is :
Sets, Relations and Function
Solution
For $1<x<2$: $[x]=1$, $f=\frac{x}{1+x^2}$ is decreasing on $x>1$, from $\frac12$ to $\frac25$: values $\left(\frac25,\frac12\right)$. For $2\le x<3$: $[x]=2$, $f=\frac{2x}{1+x^2}$ decreasing, from $\frac45$ (attained at $x=2$) to $\frac35$ (not attained): $\left(\frac35,\frac45\right]$. Range $=\left(\frac25,\frac12\right)\cup\left(\frac35,\frac45\right]$. Option (C).