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An analyst at a paper company made the table below of the sales by month at his company.

| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|-------|----|----|----|----|----|----|----|----|----|----|----|----|
| Units Sold (in thousands) | 24 | 29 | 53 | 35 | 24 | 56 | 81 | 81 | 74 | 74 | 76 | 11 |

Which of the following does his data represent?

A. Both a relation and a function
B. A function only
C. Neither a relation nor a function
D. A relation only

Sagot :

GinnyAnswer
To determine what the analyst's data represents, let's first understand the concepts of a relation and a function in the context of this sales data.

### Relation:
A relation is a set of ordered pairs, typically represented as (input, output). In this case, the input is the month number, and the output is the units sold. A table showing this data can be seen as a collection of ordered pairs: (1, 24), (2, 29), (3, 53), and so on.

### Function:
A function is a special type of relation where each input is associated with exactly one output. This means that for each month (input), there must be a single corresponding number of units sold (output).

Now, let's consider the given sales data:

| Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|--------------|----|----|----|----|----|----|----|----|----|----|----|----|
| Units Sold | 24 | 29 | 53 | 35 | 24 | 56 | 81 | 81 | 74 | 74 | 76 | 11 |

### Step-by-Step Solution:

1. Checking the Relation:
- We list out the ordered pairs: [tex]\((1, 24), (2, 29), (3, 53), (4, 35), (5, 24), (6, 56), (7, 81), (8, 81), (9, 74), (10, 74), (11, 76), (12, 11)\)[/tex].
- All month numbers have corresponding units sold quantities. Therefore, this constitutes a relation.

2. Checking the Function:
- For each month, there is a single corresponding value of units sold.
- We need to confirm there are no months that repeat with different sales values. For each month number (1 to 12), there is a unique units sold value associated with it.
- No month appears with more than one units sold value.

Since every input (month) has exactly one corresponding output (units sold), the data indeed forms both a relation and a function.

### Conclusion:
Based on the analysis, the data set represents both a relation and a function.

Therefore, the correct answer is:

A. both a relation and a function
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