Let's analyze each set of ordered pairs to determine whether it represents a function. A relation is a function if each input (x-value) maps to exactly one output (y-value). This means that for every x-value in the set, there should be only one corresponding y-value.
### Set 1: [tex]\(\{(2,-8),(6,4),(-3,9),(2,0),(-5,3)\}\)[/tex]
- The first pair is [tex]\((2, -8)\)[/tex]. The input [tex]\(2\)[/tex] maps to [tex]\(-8\)[/tex].
- The second pair is [tex]\((6, 4)\)[/tex]. The input [tex]\(6\)[/tex] maps to [tex]\(4\)[/tex].
- The third pair is [tex]\((-3, 9)\)[/tex]. The input [tex]\(-3\)[/tex] maps to [tex]\(9\)[/tex].
- The fourth pair is [tex]\((2, 0)\)[/tex]. Notice that we already have the input [tex]\(2\)[/tex] from the first pair, but now it maps to a different output [tex]\(0\)[/tex].
- The fifth pair is [tex]\((-5, 3)\)[/tex]. The input [tex]\(-5\)[/tex] maps to [tex]\(3\)[/tex].
Since the input [tex]\(2\)[/tex] maps to both [tex]\(-8\)[/tex] and [tex]\(0\)[/tex], this set of ordered pairs is not a function.
### Set 2: [tex]\(\{(1,2),(2,3),(3,4),(2,1),(1,0)\}\)[/tex]
- The first pair is [tex]\((1, 2)\)[/tex]. The input [tex]\(1\)[/tex] maps to [tex]\(2\)[/tex].
- The second pair is [tex]\((2, 3)\)[/tex]. The input [tex]\(2\)[/tex] maps to [tex]\(3\)[/tex].
- The third pair is [tex]\((3, 4)\)[/tex]. The input [tex]\(3\)[/tex] maps to [tex]\(4\)[/tex].
- The fourth pair is [tex]\((2, 1)\)[/tex]. Notice that we already have the input [tex]\(2\)[/tex] from the second pair, but now it maps to a different output [tex]\(1\)[/tex].
- The fifth pair is [tex]\((1, 0)\)[/tex]. Notice that we already have the input [tex]\(1\)[/tex] from the first pair, but now it maps to a different output [tex]\(0\)[/tex].
Since the input [tex]\(2\)[/tex] maps to both [tex]\(3\)[/tex] and [tex]\(1\)[/tex], and the input [tex]\(1\)[/tex] maps to both [tex]\(2\)[/tex] and [tex]\(0\)[/tex], this set of ordered pairs is not a function.
### Set 3: [tex]\(\{(-2,5),(7,5),(-4,0),(3,1),(0,-6)\}\)[/tex]
- The first pair is [tex]\((-2, 5)\)[/tex]. The input [tex]\(-2\)[/tex] maps to [tex]\(5\)[/tex].
- The second pair is [tex]\((7, 5)\)[/tex]. The input [tex]\(7\)[/tex] maps to [tex]\(5\)[/tex].
- The third pair is [tex]\((-4, 0)\)[/tex]. The input [tex]\(-4\)[/tex] maps to [tex]\(0\)[/tex].
- The fourth pair is [tex]\((3, 1)\)[/tex]. The input [tex]\(3\)[/tex] maps to [tex]\(1\)[/tex].
- The fifth pair is [tex]\((0, -6)\)[/tex]. The input [tex]\(0\)[/tex] maps to [tex]\(-6\)[/tex].
All x-values are unique, meaning each input has a single, unique output. Therefore, this set of ordered pairs is a function.
### Set 4: [tex]\(\{(1,-3),(1,-1),(1,1),(1,3),(1,5)\}\)[/tex]
- The first pair is [tex]\((1, -3)\)[/tex]. The input [tex]\(1\)[/tex] maps to [tex]\(-3\)[/tex].
- The second pair is [tex]\((1, -1)\)[/tex]. Notice that we already have the input [tex]\(1\)[/tex] from the first pair, but now it maps to a different output [tex]\(-1\)[/tex].
- The third pair is [tex]\((1, 1)\)[/tex]. Again, the input [tex]\(1\)[/tex] maps to yet another different output [tex]\(1\)[/tex].
- The fourth pair is [tex]\((1, 3)\)[/tex]. The input [tex]\(1\)[/tex] maps to another different output [tex]\(3\)[/tex].
- The fifth pair is [tex]\((1, 5)\)[/tex]. The input [tex]\(1\)[/tex] maps to yet another different output [tex]\(5\)[/tex].
Since the input [tex]\(1\)[/tex] maps to multiple different outputs, this set of ordered pairs is not a function.
### Summary:
- Set 1: Not a function.
- Set 2: Not a function.
- Set 3: Is a function.
- Set 4: Not a function.
Therefore, the only set that represents a function is Set 3: [tex]\(\{(-2,5),(7,5),(-4,0),(3,1),(0,-6)\}\)[/tex].
