Question

Let $f : \left(1, 3\right) \rightarrow R$ be a function defined by $f \left(x\right)=\frac{x\left[x\right]}{1+x^{2}},$ where $[x]$ denotes the greatest integer $\le x.$ Then the range of $f$ is :

• A. $\left(\frac{2}{5}, \frac{3}{5}\right)\cup \left(\frac{3}{4}, \frac{4}{5}\right)$
• B. $(\frac{2}{5}, \frac{4}{5})$
• C. $\left(\frac{3}{5}, \frac{4}{5}\right)$
• D. $\left(\frac{2}{5}, \frac{1}{2}\right) \cup (\frac{3}{5}, \frac{4}{5})$

Answer 1

Answer: (D)

$f(x) = \begin{cases} \frac{x}{x^{2}+1}\quad; & \text{$x \in\left(1, 2\right)$} \\[2ex] \frac{2x}{x^{2}+1}\quad; & \text{$x \in\left(2, 3\right)$} \end{cases}$
$f(x)$ is decreasing function
$\therefore f\left(x\right) \in\left(\frac{2}{5}, \frac{1}{2}\right) \cup \left(\frac{3}{5}, \frac{4}{5}\right)$