The function $f(x) = \lfloor x \rfloor$ represents the greatest integer less than or equal to $x$. This is commonly known as the floor function, and it maps any real number to the largest integer that is less than or equal to that number.

Answering the questions in the image:

1. Which one is the graph of $f(x) = \lfloor x \rfloor$?

   • The graph of the floor function appears as a step function where the value of $f(x)$ stays constant between consecutive integers and jumps down to the next integer at each integer value of $x$.
   • Correct answer: Option D represents the graph of the floor function, as it shows this stepwise behavior.

2. What is the function value when $x = 0.3$?

   • Since $f(x) = \lfloor x \rfloor$, for $x = 0.3$, the greatest integer less than or equal to 0.3 is 0.
   • Function value: $f(0.3) = 0$.

Do you have any questions or would you like further details on any part of this?

Here are 5 related questions to expand on the concepts:

1. How does the floor function behave when $x$ is negative, for example, $x = -0.7$?
2. What is the difference between the floor function $\lfloor x \rfloor$ and the ceiling function $\lceil x \rceil$?
3. How can the floor function be applied in real-life scenarios?
4. What is the function value $f(x) = \lfloor x \rfloor$ for $x = 1.99$?
5. How does the graph of the ceiling function compare to that of the floor function?

Tip: For fractional values, the floor function always rounds down to the nearest integer, no matter how close the number is to the next integer.