Question

Number of elements in the range set of $f(x)=\left[\frac{x}{15}\right]\left[-\frac{15}{x}\right]$, for all $x \in (0,90)$; (where $[\cdot]$ denotes the greatest integer function) is:

• A. 8
• B. 7
• C. 6
• D. 5

Hint:

Notice how the inverse term $\left[-15/x\right]$ locks into a constant value of $-1$ for all values where $x \ge 15$. This simplifies the product down to a straightforward sequence of consecutive negative integers, allowing you to find the number of elements just by counting the interval steps!

Answer 1

The Correct Option is C

Concept: To find the range set of a function involving the Greatest Integer Function $[\cdot]$, we partition the given domain $(0, 90)$ into smaller sub-intervals based on where the values inside the floor brackets step into new integers, and then evaluate the outputs. Step 1: Evaluate the function for the first segment $x \in (0, 15)$.
When $x$ lies between 0 and 15, the argument inside the first bracket sits within the fractional range $\frac{x}{15} \in (0, 1)$. Therefore, its floor value is: $$\left[\frac{x}{15}\right] = 0 \quad \Rightarrow \quad f(x) = 0 \times \left[-\frac{15}{x}\right] = 0$$ This gives us our first unique range element: $0$.

Step 2: Evaluate the function for the remaining segments where $x \ge 15$.
For all subsequent intervals, the first term $\left[\frac{x}{15}\right]$ will evaluate to a non-zero positive integer, while the second argument $-\frac{15}{x}$ will sit between $-1$ and $0$ since $x \ge 15$: $$\text{For } x \ge 15 \implies \frac{15}{x} \in (0, 1] \implies -\frac{15}{x} \in [-1, 0)$$ The floor of any number inside the half-open interval $[-1, 0)$ is always exactly $-1$. Therefore, for all values of $x \ge 15$, the second factor locks into a constant value: $\left[-\frac{15}{x}\right] = -1$.

Step 3: Calculate the outputs for each remaining interval step.
Now, multiply the stepping values of the first function by our constant factor of $-1$:
• For $x \in [15, 30)$:} $\left[\frac{x}{15}\right] = 1 \implies f(x) = 1 \times (-1) = -1$
• For $x \in [30, 45)$: $\left[\frac{x}{15}\right] = 2 \implies f(x) = 2 \times (-1) = -2$
• For $x \in [45, 60)$: $\left[\frac{x}{15}\right] = 3 \implies f(x) = 3 \times (-1) = -3$
• For $x \in [60, 75)$: $\left[\frac{x}{15}\right] = 4 \implies f(x) = 4 \times (-1) = -4$
• For $x \in [75, 90)$: $\left[\frac{x}{15}\right] = 5 \implies f(x) = 5 \times (-1) = -5$ \end{itemize

Step 4: Collect the unique values to find the size of the range set.
Gathering all the unique outputs produced across the different intervals: $$\text{Range Set } R = \{0, -1, -2, -3, -4, -5\}$$ Counting the elements shows that the set contains exactly 6 unique elements, which matches option (C).