A set of ordered pairs represents a function if and only if each element in the domain (the set of all first components of the ordered pairs) is associated with exactly one element in the range (the set of all second components). In other words, no two different ordered pairs in the set can have the same first element with different second elements.
Let’s examine each set of ordered pairs to check if it represents a function:
Set A: [tex]\(\{(1,4),(2,4),(3,4),(4,3),(4,2)\}\)[/tex]
- The domain elements are [tex]\(1, 2, 3, 4, 4\)[/tex].
- The first element [tex]\(4\)[/tex] is repeated with different second elements [tex]\((4,3)\)[/tex] and [tex]\((4,2)\)[/tex].
- Since [tex]\(4\)[/tex] is associated with both [tex]\(3\)[/tex] and [tex]\(2\)[/tex], this set does not represent a function.
Set B: [tex]\(\{(5,1),(5,2),(5,3),(5,4),(5,5)\}\)[/tex]
- The domain elements are [tex]\(5, 5, 5, 5, 5\)[/tex].
- The first element [tex]\(5\)[/tex] is repeated with different second elements [tex]\((5,1)\)[/tex], [tex]\((5,2)\)[/tex], [tex]\((5,3)\)[/tex], [tex]\((5,4)\)[/tex], and [tex]\((5,5)\)[/tex].
- Since [tex]\(5\)[/tex] is associated with multiple different values, this set does not represent a function.
Set C: [tex]\(\{(0,7),(2,7),(4,7),(6,7),(8,7)\}\)[/tex]
- The domain elements are [tex]\(0, 2, 4, 6, 8\)[/tex].
- All first elements are unique.
- Each first element is associated with exactly one second element.
- Therefore, this set represents a function.
Set D: [tex]\(\{(1,2),(2,3),(3,2),(2,1),(1,0)\}\)[/tex]
- The domain elements are [tex]\(1, 2, 3, 2, 1\)[/tex].
- The first element [tex]\(1\)[/tex] is repeated with different second elements [tex]\((1,2)\)[/tex] and [tex]\((1,0)\)[/tex], and the first element [tex]\(2\)[/tex] is associated with [tex]\((2,3)\)[/tex] and [tex]\((2,1)\)[/tex].
- Since both [tex]\(1\)[/tex] and [tex]\(2\)[/tex] are associated with multiple different values, this set does not represent a function.
Based on this analysis, the set of ordered pairs that represents a function is:
[tex]\[\boxed{C}\][/tex]
