In this problem, the function [tex]\( f(x) = \left\lfloor \frac{x}{10} \right\rfloor \)[/tex] is given to describe the number of free nights earned based on the number of nights a customer stays at a hotel.
To understand what this function indicates, let's break it down step-by-step:
1. The expression [tex]\( \frac{x}{10} \)[/tex] calculates the number of groups of 10 nights in the total number of nights stayed, [tex]\( x \)[/tex].
2. The floor function [tex]\( \left\lfloor \cdot \right\rfloor \)[/tex] takes the largest integer less than or equal to its argument. This means it rounds down to the nearest whole number.
For example:
- If a customer stays 9 nights, [tex]\( \frac{9}{10} = 0.9 \)[/tex], and [tex]\( \left\lfloor 0.9 \right\rfloor = 0 \)[/tex]. This means no free nights are earned.
- If a customer stays 10 nights, [tex]\( \frac{10}{10} = 1.0 \)[/tex], and [tex]\( \left\lfloor 1.0 \right\rfloor = 1 \)[/tex]. This means 1 free night is earned.
- If a customer stays 25 nights, [tex]\( \frac{25}{10} = 2.5 \)[/tex], and [tex]\( \left\lfloor 2.5 \right\rfloor = 2 \)[/tex]. This means 2 free nights are earned.
Thus, the function [tex]\( f(x) = \left\lfloor \frac{x}{10} \right\rfloor \)[/tex] shows that a customer earns 1 free night for every 10 nights stayed.
Therefore, the correct description of [tex]\( f(x) \)[/tex] is:
A customer earns 1 free night per 10 nights stayed.
