If the domain of the function f ( x ) = [ x ] 1 + x 2 , where [x] is greatest integer x , is [2,6), then its range is [2023]

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Q.
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 [2023]

1 $(\frac{5}{37}, \frac{2}{5}]$

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

3 $(\frac{5}{26}, \frac{2}{5}]$

4 $(\frac{5}{37}, \frac{2}{5}] - {\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}}$

Ans.
(1)

We have, $f (x) = \frac{[x]}{1 + x^{2}} and domain = [2, 6)$

$∴$ $f (x) =$ ${\begin{matrix} \frac{2}{1 + x^{2}}; & [2, 3) \\ \frac{3}{1 + x^{2}}; & [3, 4) \\ \frac{4}{1 + x^{2}}; & [4, 5) \\ \frac{5}{1 + x^{2}}; & [5, 6) \end{matrix}$

For $x \in [2, 6)$ , $f (x) > 0$ and it is a decreasing function.

At $x = 2, f (x) = \frac{2}{5}$ and at $x = 6, f (x) = \frac{5}{37}$

$Hence, Range = (\frac{5}{37}, \frac{2}{5}]$

Solution

Q.
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 [2023]

1 $(\frac{5}{37}, \frac{2}{5}]$

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

3 $(\frac{5}{26}, \frac{2}{5}]$

4 $(\frac{5}{37}, \frac{2}{5}] - {\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}}$

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Solution

Q. If the domain of the function f(x)=[x]1+x2, where [x] is greatest integer ≤x, is [2,6), then its range is [2023]

1 (537,25]

2 (526,25]-{929,27109,1889,953}

3 (526,25]

4 (537,25]-{929,27109,1889,953}

Ans. (1) We have, f(x)=[x]1+x2 and domain =[2,6) ∴ f(x)={21+x2;[2,3)31+x2;[3,4)41+x2;[4,5)51+x2;[5,6) For x∈[2,6), f(x)>0 and it is a decreasing function. At x=2, f(x)=25 and at x=6, f(x)=537 Hence, Range =(537,25]

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Q.
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 [2023]

1 $(\frac{5}{37}, \frac{2}{5}]$

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

3 $(\frac{5}{26}, \frac{2}{5}]$

4 $(\frac{5}{37}, \frac{2}{5}] - {\frac{9}{29}, \frac{27}{109}, \frac{18}{89}, \frac{9}{53}}$