To interpret the notation [tex]\( b(20) = 3 \)[/tex] within the context of the problem, we need to analyze what the function [tex]\( b \)[/tex] and its input and output stand for. According to the given table, [tex]\( b(x) \)[/tex] denotes the number of badges earned by selling [tex]\( x \)[/tex] number of boxes of cards.
Let's break down the information provided step by step:
1. We are given a series of data points that show a relationship between the number of boxes sold and the badges earned:
- Selling 1 box earns 1 badge.
- Selling 10 boxes earns 2 badges.
- Selling 20 boxes earns 3 badges.
- Selling 50 boxes earns 4 badges.
- Selling 100 boxes earns 5 badges.
- Selling 250 boxes earns 6 badges.
- Selling 500 boxes earns 7 badges.
2. The notation [tex]\( b(20) \)[/tex] implies the function [tex]\( b \)[/tex] evaluated at 20, where 20 represents the number of boxes sold.
3. The equation [tex]\( b(20) = 3 \)[/tex] directly tells us the result of this function: selling 20 boxes of cards results in earning 3 badges.
Given our understanding of the table and the notation, let's match this to the provided options:
- A. Someone who sells 3 boxes of cards earns 20 badges.
This option confuses the order of the variables and the result. It erroneously suggests a relationship that does not hold according to the table data.
- B. There are 20 people who earned 3 badges each.
This option incorrectly interprets the relationship. The function describes how many badges are earned for a certain number of boxes sold, not the number of people who earned badges.
- C. Someone who sells 20 boxes of cards earns 3 badges.
This option aligns perfectly with our interpretation from the table and the equation [tex]\( b(20) = 3 \)[/tex]. It correctly describes the situation that selling 20 boxes of cards earns 3 badges.
- D. There are 3 people who sold 20 boxes of cards each.
This option incorrectly interprets the function. It suggests a count of people, not the relationship between boxes sold and badges earned.
Thus, the correct interpretation of [tex]\( b(20) = 3 \)[/tex] in terms of the problem is:
C. Someone who sells 20 boxes of cards earns 3 badges.
