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

Engineering

Mathematics

range of function

Question

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

$(\frac{3}{4}, \frac{4}{5})$

$(\frac{2}{5}, \frac{3}{5}] \cup (\frac{3}{4}, \frac{4}{5})$

$(\frac{2}{5}, \frac{1}{2}) \cup (\frac{3}{5}, \frac{4}{5}]$

$(\frac{2}{5}, \frac{4}{5}]$

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$f (x) {\begin{matrix} \frac{x}{x^{2} + 1}; & x \in (1, 2) \\ \frac{2 x}{x^{2} + 1}; & x \in [2, 3) \end{matrix}$

$∴$ f(x) is a decreasing function

$∴ y \in (\frac{2}{5}, \frac{1}{2}) \cup (\frac{6}{10}, \frac{4}{5}]$

⇒ $y \in (\frac{2}{5}, \frac{1}{2}) \cup (\frac{3}{5}, \frac{4}{5}]$