Question

domain | range
3 | -6
9 | -1
8 | -1
-1 | -1
5 | -1
the relation is a function because select in the domain maps to select value in the range.
the relation is not a function because select in the domain maps to select value in the range.

Explanation:

Step1: Recall the definition of a function

A relation is a function if every element in the domain maps to exactly one element in the range. That is, no two different ordered pairs have the same first element (domain value) with different second elements (range values). However, multiple domain values can map to the same range value.

Step2: Analyze the given domain and range values

Looking at the table:

• Domain values: \(3, 9, 8, - 1, 5\)
• Range values: \(-6, -1, -1, -1, -1\)

We see that \(9, 8, - 1, 5\) all map to \(-1\), but each domain value (3, 9, 8, - 1, 5) is unique in the domain. Wait, no—wait, the key is: for a function, each domain element (input) has exactly one range element (output). So even if multiple inputs map to the same output, it's still a function. The only time it's not a function is if one input maps to multiple outputs.

In our case, let's check each domain element:

• \(3\) maps to \(-6\) (only one)
• \(9\) maps to \(-1\) (only one)
• \(8\) maps to \(-1\) (only one)
• \(-1\) maps to \(-1\) (only one)
• \(5\) maps to \(-1\) (only one)

So every domain element maps to exactly one range element. Now, the first option says "The relation is a function because [each element] in the domain maps to [exactly one] value in the range." The second option is incorrect because no domain element maps to more than one range value.

So filling in the blanks:

For the first (correct) option:

"The relation is a function because each element in the domain maps to exactly one value in the range."

Answer:

The relation is a function because each element in the domain maps to exactly one value in the range.

(So for the dropdowns: first "Select" in the first blank: "each element", second "Select" in the second blank: "exactly one")