A hotel offers a reward program based on the number of nights stayed. The function [tex]f(x)[/tex] represents the number of free nights earned as a function of [tex]x[/tex], the number of nights stayed.

[tex]\[ f(x) = \left\lfloor \frac{x}{10} \right\rfloor \][/tex]

Which describes the meaning of [tex]f(x)[/tex]?

A. A customer earns 1 free night per 10 nights stayed.

B. A customer starts with 1 free night and then earns another free night after every 10 nights stayed.

C. A customer earns [tex]x - 10[/tex] free nights for every 10 nights stayed.

D. A customer earns 10 free nights after [tex]x[/tex] number of nights stayed.



Answer :

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.