To determine if the given set of ordered pairs represents a function, we need to understand the definition of a function. In mathematics, a relation is a function if and only if every element in the domain (the set of all possible x-values) corresponds to exactly one element in the codomain (the set of all possible y-values). In other words, each x-value must map to one and only one y-value.
Let's analyze the given set of ordered pairs:
[tex]\[
\{(-2, 2), (-1, 2), (3, -1), (3, 1), (4, 11)\}
\][/tex]
1. List the x-values of each pair:
- [tex]\(-2\)[/tex]
- [tex]\(-1\)[/tex]
- [tex]\(3\)[/tex]
- [tex]\(3\)[/tex]
- [tex]\(4\)[/tex]
2. Check for duplicate x-values:
We see that the x-value [tex]\(3\)[/tex] appears twice: once in the ordered pair [tex]\((3, -1)\)[/tex] and once in the ordered pair [tex]\((3, 1)\)[/tex]. This means that the x-value [tex]\(3\)[/tex] corresponds to two different y-values: [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
3. Determine if it meets the definition of a function:
Because the x-value [tex]\(3\)[/tex] corresponds to two different y-values ([tex]\(-1\)[/tex] and [tex]\(1\)[/tex]), this relation does not meet the definition of a function. A function requires that each x-value maps to exactly one y-value.
Now let's address the given options:
- Option A: No, because one x-value corresponds to two different y-values.
- This is correct. We have shown that the x-value [tex]\(3\)[/tex] corresponds to both [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
- Option B: No, because two of the y-values are the same.
- This is not a valid reason. Having the same y-values does not violate the definition of a function.
- Option C: Yes, because there are two x-values that are the same.
- This is incorrect. The mere fact that x-values are the same does not mean it is a function. What matters is that each x-value corresponds to exactly one y-value.
- Option D: Yes, because every x-value corresponds to exactly one y-value.
- This is incorrect, as we’ve established that the x-value [tex]\(3\)[/tex] corresponds to two different y-values.
The correct answer is Option A: No, because one x-value corresponds to two different y-values.
