Answer :
To determine which relation represents a function, we need to understand the definition of a function. A function is a relation in which each input (x value) is associated with exactly one output (y value). In other words, no x value should map to more than one y value.
Let's analyze each of the given relations one by one:
### Option 1: [tex]\(\{(0,0),(2,3),(2,5),(6,6)\}\)[/tex]
- The input [tex]\(x = 0\)[/tex] maps to [tex]\(y = 0\)[/tex].
- The input [tex]\(x = 2\)[/tex] maps to both [tex]\(y = 3\)[/tex] and [tex]\(y = 5\)[/tex].
- The input [tex]\(x = 6\)[/tex] maps to [tex]\(y = 6\)[/tex].
Since the input [tex]\(x = 2\)[/tex] maps to two different outputs ([tex]\(y = 3\)[/tex] and [tex]\(y = 5\)[/tex]), this relation does not represent a function.
### Option 2: [tex]\(\{(3,5),(8,4),(10,11),(10,6)\}\)[/tex]
- The input [tex]\(x = 3\)[/tex] maps to [tex]\(y = 5\)[/tex].
- The input [tex]\(x = 8\)[/tex] maps to [tex]\(y = 4\)[/tex].
- The input [tex]\(x = 10\)[/tex] maps to both [tex]\(y = 11\)[/tex] and [tex]\(y = 6\)[/tex].
Since the input [tex]\(x = 10\)[/tex] maps to two different outputs ([tex]\(y = 11\)[/tex] and [tex]\(y = 6\)[/tex]), this relation does not represent a function.
### Option 3: [tex]\(\{(-2,2),(0,2),(7,2),(11,2)\}\)[/tex]
- The input [tex]\(x = -2\)[/tex] maps to [tex]\(y = 2\)[/tex].
- The input [tex]\(x = 0\)[/tex] maps to [tex]\(y = 2\)[/tex].
- The input [tex]\(x = 7\)[/tex] maps to [tex]\(y = 2\)[/tex].
- The input [tex]\(x = 11\)[/tex] maps to [tex]\(y = 2\)[/tex].
Each input [tex]\(x\)[/tex] maps to exactly one output [tex]\(y\)[/tex], which means that this relation does represent a function.
### Option 4: [tex]\(\{(13,2),(13,3),(13,4),(13,5)\}\)[/tex]
- The input [tex]\(x = 13\)[/tex] maps to [tex]\(y = 2\)[/tex].
- The input [tex]\(x = 13\)[/tex] maps to [tex]\(y = 3\)[/tex].
- The input [tex]\(x = 13\)[/tex] maps to [tex]\(y = 4\)[/tex].
- The input [tex]\(x = 13\)[/tex] maps to [tex]\(y = 5\)[/tex].
Since the input [tex]\(x = 13\)[/tex] maps to multiple outputs ([tex]\(y = 2\)[/tex], [tex]\(y = 3\)[/tex], [tex]\(y = 4\)[/tex], and [tex]\(y = 5\)[/tex]), this relation does not represent a function.
After analyzing all the options, we conclude that the relation that represents a function is:
[tex]\[ \{(-2,2),(0,2),(7,2),(11,2)\} \][/tex]
Thus, the correct answer is:
[tex]\[ 3 \][/tex]
Let's analyze each of the given relations one by one:
### Option 1: [tex]\(\{(0,0),(2,3),(2,5),(6,6)\}\)[/tex]
- The input [tex]\(x = 0\)[/tex] maps to [tex]\(y = 0\)[/tex].
- The input [tex]\(x = 2\)[/tex] maps to both [tex]\(y = 3\)[/tex] and [tex]\(y = 5\)[/tex].
- The input [tex]\(x = 6\)[/tex] maps to [tex]\(y = 6\)[/tex].
Since the input [tex]\(x = 2\)[/tex] maps to two different outputs ([tex]\(y = 3\)[/tex] and [tex]\(y = 5\)[/tex]), this relation does not represent a function.
### Option 2: [tex]\(\{(3,5),(8,4),(10,11),(10,6)\}\)[/tex]
- The input [tex]\(x = 3\)[/tex] maps to [tex]\(y = 5\)[/tex].
- The input [tex]\(x = 8\)[/tex] maps to [tex]\(y = 4\)[/tex].
- The input [tex]\(x = 10\)[/tex] maps to both [tex]\(y = 11\)[/tex] and [tex]\(y = 6\)[/tex].
Since the input [tex]\(x = 10\)[/tex] maps to two different outputs ([tex]\(y = 11\)[/tex] and [tex]\(y = 6\)[/tex]), this relation does not represent a function.
### Option 3: [tex]\(\{(-2,2),(0,2),(7,2),(11,2)\}\)[/tex]
- The input [tex]\(x = -2\)[/tex] maps to [tex]\(y = 2\)[/tex].
- The input [tex]\(x = 0\)[/tex] maps to [tex]\(y = 2\)[/tex].
- The input [tex]\(x = 7\)[/tex] maps to [tex]\(y = 2\)[/tex].
- The input [tex]\(x = 11\)[/tex] maps to [tex]\(y = 2\)[/tex].
Each input [tex]\(x\)[/tex] maps to exactly one output [tex]\(y\)[/tex], which means that this relation does represent a function.
### Option 4: [tex]\(\{(13,2),(13,3),(13,4),(13,5)\}\)[/tex]
- The input [tex]\(x = 13\)[/tex] maps to [tex]\(y = 2\)[/tex].
- The input [tex]\(x = 13\)[/tex] maps to [tex]\(y = 3\)[/tex].
- The input [tex]\(x = 13\)[/tex] maps to [tex]\(y = 4\)[/tex].
- The input [tex]\(x = 13\)[/tex] maps to [tex]\(y = 5\)[/tex].
Since the input [tex]\(x = 13\)[/tex] maps to multiple outputs ([tex]\(y = 2\)[/tex], [tex]\(y = 3\)[/tex], [tex]\(y = 4\)[/tex], and [tex]\(y = 5\)[/tex]), this relation does not represent a function.
After analyzing all the options, we conclude that the relation that represents a function is:
[tex]\[ \{(-2,2),(0,2),(7,2),(11,2)\} \][/tex]
Thus, the correct answer is:
[tex]\[ 3 \][/tex]
