If the domain of the function [f(x)=[x]1+x^2], where [[x]] is greatest integer [ x], is [[…

Question

If the domain of the function $$f(x)=\frac{[x]}{1+x^{2}}$$, where $$[x]$$ is greatest integer $$\leq x$$, is $$[2,6)$$, then its range is

Accepted Answer

$f(x)=\frac{k}{1+x^{2}}$ is a decreasing function where $k>0$

$$
\begin{gathered}
\therefore \quad x \in[2,3) \Rightarrow f(x)=\frac{2}{1+x^{2}} \in\left(\frac{2}{10}, \frac{2}{5}\right]=R_{1} \\
x \in[3,4) \Rightarrow f(x)=\frac{3}{1+x^{2}} \in\left(\frac{3}{17}, \frac{3}{10}\right]=R_{2} \\
x \in[4,5) \Rightarrow f(x)=\frac{4}{1+x^{2}} \in\left(\frac{4}{26}, \frac{4}{17}\right]=R_{3} \\
x \in[5,6) \Rightarrow f(x)=\frac{5}{1+x^{2}} \in\left(\frac{5}{37}, \frac{5}{26}\right]=R_{4} \\
\text { Range }=R_{1} \cup R_{2} \cup R_{3} \cup R_{4} \\
=\left(\frac{5}{37}, \frac{2}{5}\right]
\end{gathered}
$$

Solutions · 1