To determine which of the given relations is a function, we need to verify if each relation satisfies the definition of a function. A relation is a function if every element in the domain is associated with exactly one element in the range. This means that no two ordered pairs in the relation should have the same first element with different second elements.
Let's analyze each relation step-by-step:
1. [tex]\(\{(-4,3),(-2,-1),(-4,8)\}\)[/tex]
- The domain elements are: [tex]\(-4, -2, -4\)[/tex]
- The range elements are: [tex]\(3, -1, 8\)[/tex]
We see that the first element [tex]\(-4\)[/tex] is paired with both [tex]\(3\)[/tex] and [tex]\(8\)[/tex]. Since [tex]\(-4\)[/tex] is associated with more than one value, this relation is not a function.
2. [tex]\(\{(-4,3),(-2,-1),(-7,8)\}\)[/tex]
- The domain elements are: [tex]\(-4, -2, -7\)[/tex]
- The range elements are: [tex]\(3, -1, 8\)[/tex]
Here, each domain element [tex]\(-4, -2, -7\)[/tex] is associated with exactly one unique range element [tex]\(3, -1, 8\)[/tex], respectively. Since there are no repeated domain elements with different range elements, this relation is a function.
3. [tex]\(\{-4,-2,-7,7\}\)[/tex]
- This is a set of numbers and not pairs, hence it cannot represent a function or a relation in the traditional sense where we consider ordered pairs [tex]\((x, y)\)[/tex].
4. [tex]\(\{(-4,3),(-2,-1),(-2,-8),(-7,8)\}\)[/tex]
- The domain elements are: [tex]\(-4, -2, -2, -7\)[/tex]
- The range elements are: [tex]\(3, -1, -8, 8\)[/tex]
We see that the first element [tex]\(-2\)[/tex] is paired with both [tex]\(-1\)[/tex] and [tex]\(-8\)[/tex]. Since [tex]\(-2\)[/tex] is associated with more than one value, this relation is not a function.
In conclusion:
- [tex]\(\{(-4,3),(-2,-1),(-4,8)\}\)[/tex] is not a function.
- [tex]\(\{(-4,3),(-2,-1),(-7,8)\}\)[/tex] is a function.
- [tex]\(\{-4,-2,-7,7\}\)[/tex] cannot be considered as a function as it does not follow the ordered pair structure.
- [tex]\(\{(-4,3),(-2,-1),(-2,-8),(-7,8)\}\)[/tex] is not a function.
