The function given is [tex]\( f(x) = \left\lfloor \frac{x}{10} \right\rfloor \)[/tex], where [tex]\( x \)[/tex] is the number of nights stayed and [tex]\( f(x) \)[/tex] is the number of free nights earned. The floor function, [tex]\( \left\lfloor \cdot \right\rfloor \)[/tex], rounds down to the nearest integer.
To understand this function, let's analyze step-by-step:
1. Input Variable [tex]\(x\)[/tex]: This represents the total number of nights a customer has stayed at the hotel.
2. Division by 10: The expression [tex]\( \frac{x}{10} \)[/tex] calculates the exact number of groups of 10 nights within the total [tex]\( x \)[/tex] nights.
3. Floor Function: The floor function [tex]\( \left\lfloor \frac{x}{10} \right\rfloor \)[/tex] rounds down the result of the division to the nearest whole number (integer). This means that any fractional part is discarded.
For example:
- If a customer stays for 20 nights ([tex]\(x = 20\)[/tex]):
[tex]\[
f(20) = \left\lfloor \frac{20}{10} \right\rfloor = \left\lfloor 2 \right\rfloor = 2
\][/tex]
- If a customer stays for 25 nights ([tex]\(x = 25\)[/tex]):
[tex]\[
f(25) = \left\lfloor \frac{25}{10} \right\rfloor = \left\lfloor 2.5 \right\rfloor = 2
\][/tex]
- If a customer stays for 9 nights ([tex]\(x = 9\)[/tex]):
[tex]\[
f(9) = \left\lfloor \frac{9}{10} \right\rfloor = \left\lfloor 0.9 \right\rfloor = 0
\][/tex]
From the calculations, we see that for every complete set of 10 nights stayed, the customer earns 1 free night.
Thus, the appropriate description is: A customer earns 1 free night per 10 nights stayed.
So, the correct choice is:
- A customer earns 1 free night per 10 nights stayed.
The numerical result [tex]\( (1, 1) \)[/tex] confirms this interpretation, as it shows the output and the correct description corresponding to the question.
