We need to find the number of elements in the range set of the function \(f(x) = \left[ \frac{x}{15} \right] \left[ -\frac{15}{x} \right]\) for \(x \in (0, 90)\).
Let's break down the function \(f(x)\):
- \(\left[ \frac{x}{15} \right]\) means the greatest integer less than or equal to \(\frac{x}{15}\).
- \(\left[ -\frac{15}{x} \right]\) means the greatest integer less than or equal to \(-\frac{15}{x}\).
Let's analyze each component:
- For \(\left[ \frac{x}{15} \right]\):
- As \(x\) ranges from 0 to 90, the expression \(\frac{x}{15}\) will take values from just above 0 to just below 6.
- So, \(\left[ \frac{x}{15} \right]\) can be \(0, 1, 2, 3, 4, 5\).
- For \(\left[ -\frac{15}{x} \right]\):
- As \(x\) goes from 0 to 90, \(-\frac{15}{x}\) will range from \(-\infty\) to just less than 0 (approaching 0).
- Therefore, \(\left[ -\frac{15}{x} \right]\) can be \(-15, -14, ..., -1\) (14 elements in total).
The function \(f(x) = \left[\frac{x}{15}\right]\left[-\frac{15}{x}\right]\) is thus a product of possible integers for each component.
Let's consider values which are possible for \(f(x)\):
| \(\left[\frac{x}{15}\right]\) | \(\left[-\frac{15}{x}\right]\) | \(\left[\frac{x}{15}\right] \left[-\frac{15}{x}\right]\;\) |
|---|---|---|
| 0 | -1 | 0 |
| 1 | -1, -2 | -1, -2 |
| 2 | -1, -2, -3, -4 | -2, -3, -4, -6 |
| 3 | -1, -2, -3, -4, -5 | -3, -4, -6, -9, -12 |
| 4 | -1, -2, -3, -4 | -4, -6, -8, -12 |
| 5 | -1, -2, -3 | -5, -10, -15 |
From this, possible values for \(f(x)\) are: \(0, -1, -2, -3, -4, -5, -6, -9\).
Counting distinct values gives us \(8\) possible values.
Therefore, the number of elements in the range set of \(f(x)\) is \(8\).
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