Answer :
To understand the meaning of the function \( f(x) = \left\lfloor \frac{x}{10} \right\rfloor \), let’s break down what the function represents and then interpret it step-by-step.
1. Understanding the Function \( \left\lfloor \frac{x}{10} \right\rfloor \):
- The function \( \frac{x}{10} \) divides the number of nights stayed (\( x \)) by 10.
- The floor function, denoted as \( \left\lfloor \cdot \right\rfloor \), rounds down the result to the nearest integer.
- This means \( f(x) \) counts the number of complete sets of 10 nights stayed.
2. Interpreting the Result:
- For every 10 nights stayed, the customer earns one complete free night.
- If a customer stays for 9 nights, \( f(9) = \left\lfloor \frac{9}{10} \right\rfloor = 0 \) free nights.
- If a customer stays for 10 nights, \( f(10) = \left\lfloor \frac{10}{10} \right\rfloor = 1 \) free night.
- This pattern continues such that a customer staying for any multiple of 10 nights earns that many free nights. For example, for 20 nights, \( f(20) = \left\lfloor \frac{20}{10} \right\rfloor = 2 \) free nights, and so on.
3. Determining the Correct Description:
- From the interpretation above, we can derive the correct description of [tex]$f(x)$[/tex]:
- "A customer earns 1 free night per 10 nights stayed."
Thus, the function \( f(x) \) correctly describes that for each set of 10 nights a customer stays, they earn 1 free night.
Therefore, the correct answer is:
- A customer earns 1 free night per 10 nights stayed.
1. Understanding the Function \( \left\lfloor \frac{x}{10} \right\rfloor \):
- The function \( \frac{x}{10} \) divides the number of nights stayed (\( x \)) by 10.
- The floor function, denoted as \( \left\lfloor \cdot \right\rfloor \), rounds down the result to the nearest integer.
- This means \( f(x) \) counts the number of complete sets of 10 nights stayed.
2. Interpreting the Result:
- For every 10 nights stayed, the customer earns one complete free night.
- If a customer stays for 9 nights, \( f(9) = \left\lfloor \frac{9}{10} \right\rfloor = 0 \) free nights.
- If a customer stays for 10 nights, \( f(10) = \left\lfloor \frac{10}{10} \right\rfloor = 1 \) free night.
- This pattern continues such that a customer staying for any multiple of 10 nights earns that many free nights. For example, for 20 nights, \( f(20) = \left\lfloor \frac{20}{10} \right\rfloor = 2 \) free nights, and so on.
3. Determining the Correct Description:
- From the interpretation above, we can derive the correct description of [tex]$f(x)$[/tex]:
- "A customer earns 1 free night per 10 nights stayed."
Thus, the function \( f(x) \) correctly describes that for each set of 10 nights a customer stays, they earn 1 free night.
Therefore, the correct answer is:
- A customer earns 1 free night per 10 nights stayed.
