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  • If the domain of the function  <img alt="f(x)=\frac{\left [ a \right ]}{1+x^{2}}," src="https://entrancecorner.oncodecogs.com/gif.latex?f%28x%29%3D%5Cfrac%7B%5Cleft%20%5B%20a%20%5Cright%2

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

 

Option: 1

\left(\frac{5}{26}, \frac{2}{5}\right]


Option: 2

\left(\frac{5}{37}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}


Option: 3

\left(\frac{5}{37}, \frac{2}{5}\right]


Option: 4

\left(\frac{5}{26}, \frac{2}{5}\right]-\left\{\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}\right\}


Answers (1)

f(x)=\frac{[x]}{1+x^2}, \quad x=\in[2,6]

\begin{aligned} & f(x)= \begin{cases}\frac{2}{1+\mathrm{x}^2} & \mathrm{x} \in[2,3) \\ \frac{3}{1+\mathrm{x}^2} & \mathrm{x} \in[3,4) \\ \frac{4}{1+\mathrm{x}^2} & \mathrm{x} \in[4,5) \\ \frac{5}{1+\mathrm{x}^2} & \mathrm{x} \in[5,6)\end{cases} \\ & \because f(x) \text { is } \downarrow \text { in } x \in[2,6) \end{aligned}

range is  \left(\frac{5}{37}, \frac{2}{5}\right]

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Kshitij

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