Finding the Range of a Linear Contextual Function
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Rafi and Kayla are moving into their new home. In order to manage their time, they decide to focus on unpacking only 4 boxes each day. As shown in the graph, $B(n)$ gives the number of boxes remaining to unpack as a function of the number of days since they moved in. In this context, what is the range of $B(n)$? A) all whole numbers less than or equal to 8 B) all multiples of 4 less than or equal to 32 C) all whole numbers less than or equal to 32 D) all real numbers less than or equal to 8
This question includes visual content: A scatter plot titled 'Boxes to unpack' shows 'Days' on the x-axis ranging from 0 to 8 and 'Boxes remaining' on the y-axis ranging from 0 to 36. The graph shows discrete data points starting at $(0, 32)$ and decreasing by 4 units for each step in days, finishing at $(8, 0)$.
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Step by Step Written Solution
Rafi and Kayla are unpacking boxes at a rate of four boxes per day. We are given a graph showing the number of boxes remaining, B of n, as a function of the number of days, n. Our goal is to determine the range of this function in this context.
Identifying the Range
Recall that in a function, the range is the set of all possible output values, or the y-values. In this problem, the y-axis represents the number of boxes remaining.
Let's look closely at the graph to find the vertical coordinates of each point plotted. Since they unpack exactly four boxes each day, the values are discrete points, not a continuous line.
Analyzing Coordinates
| Day (n) | Boxes Remaining B(n) |
|---|---|
| 0 | 32 |
| 1 | 28 |
| 2 | 24 |
| 3 | 20 |
| 4 | 16 |
| 5 | 12 |
| 6 | 8 |
| 7 | 4 |
| 8 | 0 |
If we list out these output values, we have thirty-two, twenty-eight, twenty-four, and so on, down to zero.
Notice a pattern here. Every value in this set is a multiple of four.
Additionally, the highest value is thirty-two, and the lowest is zero. These are the possible total counts of remaining boxes.
Points are discrete (whole numbers of boxes).
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