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The image shows a mathematical problem asking to fill in the blank related to the greatest integer function, which is defined as producing the greatest integer *less than or equal to* \( x \).

In the equation provided:
\[
f(x) = \lfloor x \rfloor
\]
The symbol \(\lfloor x \rfloor\) represents the greatest integer function (also known as the floor function). The correct word to fill in the blank is **"less than"**.

So, the complete sentence would be:
"The greatest integer function, shown below, is defined so that it produces the greatest integer **less than** or equal to \( x \)."

Would you like more details or have any other questions?

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Here are 8 related questions you might be interested in:
1. What is the difference between the floor function and the ceiling function?
2. How is the floor function used in real-life applications?
3. Can the greatest integer function ever produce a value greater than \( x \)?
4. What is the output of the floor function for negative numbers?
5. How does the floor function differ from rounding down?
6. How would you represent the floor function graphically?
7. Can the greatest integer function be defined for complex numbers?
8. What are some common mistakes students make when working with the floor function?

**Tip:** Always remember that the floor function outputs an integer, which is the greatest integer less than or equal to the given number, meaning it rounds down to the nearest integer if the number is not already an integer.

Explore the definition and application of the greatest integer function (floor function) in mathematics. This problem involves understanding how the function produces the greatest integer less than or equal to a given number. Suitable for advanced high school students.

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