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 (1)
The function f ( x ) = ⌊ 10 x ⌋ gives the number of free nights earned based on the number of nights stayed, x .
If x = 10 , f ( 10 ) = 1 ; if x = 20 , f ( 20 ) = 2 ; if x = 9 , f ( 9 ) = 0 ; if x = 11 , f ( 11 ) = 1 .
This means that for every 10 nights stayed, the customer earns 1 free night.
Therefore, the correct answer is: A customer earns 1 free night per 10 nights stayed. A customer earns 1 free night per 10 nights stayed.
Explanation
Understanding the Function We are given the function f ( x ) = ⌊ 10 x ⌋ , which represents the number of free nights earned at a hotel based on the number of nights stayed, x . We need to determine the correct interpretation of this function from the given options.
Analyzing the Function's Behavior Let's analyze the function for a few values of x to understand its behavior.
If x = 10 , then f ( 10 ) = ⌊ 10 10 ⌋ = ⌊ 1 ⌋ = 1 . This means that after staying 10 nights, a customer earns 1 free night.
If x = 20 , then f ( 20 ) = ⌊ 10 20 ⌋ = ⌊ 2 ⌋ = 2 . This means that after staying 20 nights, a customer earns 2 free nights.
If x = 9 , then f ( 9 ) = ⌊ 10 9 ⌋ = ⌊ 0.9 ⌋ = 0 . This means that after staying 9 nights, a customer earns 0 free nights.
If x = 11 , then f ( 11 ) = ⌊ 10 11 ⌋ = ⌊ 1.1 ⌋ = 1 . This means that after staying 11 nights, a customer earns 1 free night.
Evaluating the Options Now, let's evaluate the given options:
A customer earns 1 free night per 10 nights stayed. This option aligns with our analysis. For every 10 nights, the customer earns 1 free night. This seems to be the correct answer.
A customer starts with 1 free night and then earns another free night after every 10 nights stayed. This is incorrect because f ( 0 ) = ⌊ 10 0 ⌋ = 0 , meaning the customer starts with 0 free nights.
A customer earns x − 10 free nights for every 10 nights stayed. This is incorrect. For example, if x = 10 , the customer would earn 10 − 10 = 0 free nights, but we know they earn 1.
A customer earns 10 free nights after x number of nights stayed. This is incorrect. For example, if x = 10 , the customer earns 1 free night, not 10.
Conclusion Based on our analysis, the correct description of the function f ( x ) is that a customer earns 1 free night per 10 nights stayed.
Examples
Imagine a library offers a similar reward. For every 10 books you read, you get a free rental. The function f ( x ) = ⌊ 10 x ⌋ tells you how many free rentals you've earned based on the number of books, x , you've read. This kind of reward system encourages continued engagement and provides a tangible benefit for consistent participation, making reading more rewarding.
