To determine which of the given sets of ordered pairs does not represent a function, we need to review the definition of a function. A relation is a function if and only if every input (or domain value) is associated with exactly one output (or range value). In other words, no input value (x-value) should be repeated with different output values (y-values).
Let's analyze each relation one by one:
1. Relation 1:
[tex]\[\{(5, 2), (4, 2), (3, 2), (2, 2), (1, 2)\}\][/tex]
- The x-values are [tex]\(\{5, 4, 3, 2, 1\}\)[/tex].
- Each x-value is unique, and no x-value is repeated.
Therefore, the first relation is a function.
2. Relation 2:
[tex]\[\{(-8, -3), (-6, -5), (-4, -2), (-2, -7), (-1, -4)\}\][/tex]
- The x-values are [tex]\(\{-8, -6, -4, -2, -1\}\)[/tex].
- Each x-value is unique, and no x-value is repeated.
Therefore, the second relation is a function.
3. Relation 3:
[tex]\[\{(-6, 4), (-3, -1), (0, 5), (1, -1), (2, 3)\}\][/tex]
- The x-values are [tex]\(\{-6, -3, 0, 1, 2\}\)[/tex].
- Each x-value is unique, and no x-value is repeated.
Therefore, the third relation is a function.
4. Relation 4:
[tex]\[\{(-4, -2), (-1, -1), (3, 2), (3, 5), (7, 10)\}\][/tex]
- The x-values are [tex]\(\{-4, -1, 3, 7\}\)[/tex].
- Here, the x-value [tex]\(3\)[/tex] is repeated with different y-values, specifically [tex]\( (3, 2) \)[/tex] and [tex]\( (3, 5) \)[/tex].
Since the x-value [tex]\(3\)[/tex] is associated with two different y-values ([tex]\(2\)[/tex] and [tex]\(5\)[/tex]), this violates the definition of a function.
Thus, the relation that is not a function is the fourth one:
[tex]\[\{(-4, -2), (-1, -1), (3, 2), (3, 5), (7, 10)\}\][/tex]
So, the relation set 4 is not a function.
