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

Given:

$f\left(x\right) = \dfrac{\left[x\right]}{1 + {x}^{2}}$

So,

$\begin{cases}f\left(x\right) = \dfrac{2}{1 + {x}^{2}}; & & x \in [ 2, 3 ) \\ f\left(x\right) = \dfrac{3}{1 + {x}^{2}}; & & x \in [ 3, 4 ) \\ f\left(x\right) = \dfrac{4}{1 + {x}^{2}}; & & x \in [ 4, 5 ) \\ f\left(x\right) = \dfrac{5}{1 + {x}^{2}}; & & x \in [ 5, 6 )\end{cases}$

Hence,

$f\left(x\right) \in \left(\dfrac{5}{37}, \dfrac{2}{5}\right]$