The question displayed in the image asks to evaluate $f(6.51)$ if $f(x) = \lfloor x \rfloor$, where $\lfloor x \rfloor$ represents the greatest integer less than or equal to $x$.

Step-by-Step Solution:

1. Definition of $\lfloor x \rfloor$:

   • $\lfloor x \rfloor$ takes a real number $x$ and rounds it down to the largest integer less than or equal to $x$.

2. Applying $\lfloor 6.51 \rfloor$:

   • The number $6.51$ lies between 6 and 7. The greatest integer less than or equal to $6.51$ is $6$.

3. Conclusion:

   • $f(6.51) = \lfloor 6.51 \rfloor = 6$.

Answer:

The correct choice is 6.

Would you like a further explanation or have additional questions?

Here are 5 related questions to expand your understanding:

1. What is $\lfloor -3.7 \rfloor$ using the floor function?
2. How is the floor function different from the ceiling function $\lceil x \rceil$?
3. What is $f(x)$ if $f(x) = \lfloor x + 0.5 \rfloor$?
4. Can the floor function be applied to irrational numbers such as $\sqrt{2}$? If so, how?
5. How can the floor function be used in programming to round down numbers?

Tip: Remember that the floor function always rounds down, even for negative numbers. For example, $\lfloor -2.3 \rfloor = -3$.