To determine whether the given relation is a function, we need to check if each input (or [tex]\(x\)[/tex]-value) is associated with exactly one output (or [tex]\(y\)[/tex]-value). In other words, in a function, no [tex]\(x\)[/tex]-value should map to more than one [tex]\(y\)[/tex]-value.
The given set of ordered pairs is:
[tex]\[
(3,-7), (2,4), (-3,-7), (3,0), (0,2)
\][/tex]
We will examine each [tex]\(x\)[/tex]-value in the pairs and see if any [tex]\(x\)[/tex]-value is repeated with a different [tex]\(y\)[/tex]-value.
1. The first pair is [tex]\((3, -7)\)[/tex].
2. The second pair is [tex]\((2, 4)\)[/tex].
3. The third pair is [tex]\((-3, -7)\)[/tex].
4. The fourth pair is [tex]\((3, 0)\)[/tex].
Here, we can see that the [tex]\(x\)[/tex]-value [tex]\(3\)[/tex] appears again. However, it is associated with a different [tex]\(y\)[/tex]-value:
- In the first pair: [tex]\( (3, -7) \)[/tex]
- In the fourth pair: [tex]\( (3, 0) \)[/tex]
Since the [tex]\(x\)[/tex]-value [tex]\(3\)[/tex] is paired with two different [tex]\(y\)[/tex]-values ([tex]\(-7\)[/tex] and [tex]\(0\)[/tex]), this indicates that one input maps to multiple outputs.
Therefore, the relation given by the set of ordered pairs is not a function.
The answer is:
[tex]\[
\text{IN}
\][/tex]
