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
Solution
$f(x)=\frac{k}{1+x^{2}}$ is a decreasing function where $k>0$
<br/><br/>$$
\begin{gathered}
\therefore \quad x \in[2,3) \Rightarrow f(x)=\frac{2}{1+x^{2}} \in\left(\frac{2}{10}, \frac{2}{5}\right]=R_{1} \\\\
x \in[3,4) \Rightarrow f(x)=\frac{3}{1+x^{2}} \in\left(\frac{3}{17}, \frac{3}{10}\right]=R_{2} \\\\
x \in[4,5) \Rightarrow f(x)=\frac{4}{1+x^{2}} \in\left(\frac{4}{26}, \frac{4}{17}\right]=R_{3} \\\\
x \in[5,6) \Rightarrow f(x)=\frac{5}{1+x^{2}} \in\left(\frac{5}{37}, \frac{5}{26}\right]=R_{4} \\\\
\text { Range }=R_{1} \cup R_{2} \cup R_{3} \cup R_{4} \\\\
=\left(\frac{5}{37}, \frac{2}{5}\right]
\end{gathered}
$$
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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