Sets, Relations & Functions
Range with Greatest Integer Function
nta_pyq_2023_jan
Grade None
Question:
If the domain of the function $f(x) = \dfrac{[x]}{1+x^2}$, where $[x]$ is greatest integer $\le x$, is $(2, 6)$, then its range is
$\left(\dfrac{5}{26}, \dfrac{2}{5}\right) - \left\{\dfrac{9}{29}, \dfrac{27}{109}, \dfrac{18}{89}, \dfrac{9}{53}\right\}$
$\left(\dfrac{5}{26}, \dfrac{2}{5}\right]$
$\left(\dfrac{5}{37}, \dfrac{2}{5}\right) - \left\{\dfrac{9}{29}, \dfrac{27}{109}, \dfrac{18}{89}, \dfrac{9}{53}\right\}$
$\left(\dfrac{5}{37}, \dfrac{2}{5}\right)$
Step-by-Step Solution
Key Concept: For $x \in (2,6)$, $[x] \in \{2,3,4,5\}$; compute the range on each integer interval and combine.
On each subinterval $[n, n+1)$, $f(x)=n/(1+x^2)$, which is continuous and decreasing. Combined range is $\left(\frac{5}{37}, \frac{2}{5}\right)$ with excluded points at integer $x$ values.
Correct Answer: 4