Question

Let $f:\left(1,3\right)\to R$ 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)

$f(x)=\{\begin{array}{ll}\frac{x}{x^2+1};& \text{x∈(1,2)}\\ \frac{2x}{x^2+1};& \text{x∈(2,3)}\end{array}$
$f(x)$ is decreasing function
$\therefore f\left(x\right)\in \left(\frac{2}{5},\frac{1}{2}\right)\cup \left(\frac{3}{5},\frac{4}{5}\right)$