To determine which of the given relations is a function, we need to verify that each relation satisfies the definition of a mathematical function. Specifically, a relation is a function if every input value (x-value) corresponds to exactly one output value (y-value). In other words, for each x in the set of ordered pairs, there must be only one y associated with it.
Let's examine each set of ordered pairs one by one:
1. [tex]\(\{(-3, -2), (-2, -1), (0, -1), (0, 1), (1, 2)\}\)[/tex]
- The ordered pairs are: [tex]\((-3, -2)\)[/tex], [tex]\((-2, -1)\)[/tex], [tex]\((0, -1)\)[/tex], [tex]\((0, 1)\)[/tex], [tex]\((1, 2)\)[/tex].
- Here, the x-value 0 is associated with two different y-values ([tex]\(-1\)[/tex] and [tex]\(1\)[/tex]).
- Since the x-value 0 has multiple y-values, this relation is not a function.
2. [tex]\(\{(5, 4), (5, 6), (5, 8), (5, 10), (5, 12)\}\)[/tex]
- The ordered pairs are: [tex]\((5, 4)\)[/tex], [tex]\((5, 6)\)[/tex], [tex]\((5, 8)\)[/tex], [tex]\((5, 10)\)[/tex], [tex]\((5, 12)\)[/tex].
- Here, the x-value 5 is associated with multiple y-values (4, 6, 8, 10, and 12).
- Since the x-value 5 has multiple y-values, this relation is not a function.
3. [tex]\(\{(7, 3), (-6, 8), (-3, 5), (0, -3), (7, 11)\}\)[/tex]
- The ordered pairs are: [tex]\((7, 3)\)[/tex], [tex]\((-6, 8)\)[/tex], [tex]\((-3, 5)\)[/tex], [tex]\((0, -3)\)[/tex], [tex]\((7, 11)\)[/tex].
- Here, the x-value 7 is associated with two different y-values (3 and 11).
- Since the x-value 7 has multiple y-values, this relation is not a function.
4. [tex]\(\{(5, 2), (-4, 2), (3, 6), (0, 4), (-1, 2)\}\)[/tex]
- The ordered pairs are: [tex]\((5, 2)\)[/tex], [tex]\((-4, 2)\)[/tex], [tex]\((3, 6)\)[/tex], [tex]\((0, 4)\)[/tex], [tex]\((-1, 2)\)[/tex].
- Each x-value maps to exactly one y-value: 5 to 2, -4 to 2, 3 to 6, 0 to 4, and -1 to 2.
- Since each x-value has a unique y-value, this relation is a function.
In conclusion, the only relation that is a function is:
[tex]\[
\{(5, 2), (-4, 2), (3, 6), (0, 4), (-1, 2)\}
\][/tex]
