To determine which relation represents a function, we need to recall the definition of a function. A relation is a function if each input (or [tex]\( x \)[/tex]-value) is associated with exactly one output (or [tex]\( y \)[/tex]-value). In other words, no [tex]\( x \)[/tex]-value should be associated with more than one [tex]\( y \)[/tex]-value.
Let's analyze each given relation:
1. Relation 1: [tex]\(\{(0,0),(2,3),(2,5),(6,6)\}\)[/tex]
- For [tex]\( x = 0 \)[/tex], the corresponding [tex]\( y \)[/tex]-value is [tex]\( 0 \)[/tex].
- For [tex]\( x = 2 \)[/tex], there are two corresponding [tex]\( y \)[/tex]-values: [tex]\( 3 \)[/tex] and [tex]\( 5 \)[/tex].
Since the [tex]\( x \)[/tex]-value [tex]\( 2 \)[/tex] is associated with more than one [tex]\( y \)[/tex]-value, this relation does not represent a function.
2. Relation 2: [tex]\(\{(3,5),(8,4),(10,11),(10,8)\}\)[/tex]
- For [tex]\( x = 3 \)[/tex], the corresponding [tex]\( y \)[/tex]-value is [tex]\( 5 \)[/tex].
- For [tex]\( x = 8 \)[/tex], the corresponding [tex]\( y \)[/tex]-value is [tex]\( 4 \)[/tex].
- For [tex]\( x = 10 \)[/tex], there are two corresponding [tex]\( y \)[/tex]-values: [tex]\( 11 \)[/tex] and [tex]\( 8 \)[/tex].
Since the [tex]\( x \)[/tex]-value [tex]\( 10 \)[/tex] is associated with more than one [tex]\( y \)[/tex]-value, this relation does not represent a function.
3. Relation 3: [tex]\(\{(-2,2),(0,2),(-2,1),(2,2)\}\)[/tex]
- For [tex]\( x = -2 \)[/tex], there are two corresponding [tex]\( y \)[/tex]-values: [tex]\( 2 \)[/tex] and [tex]\( 1 \)[/tex].
- For [tex]\( x = 0 \)[/tex], the corresponding [tex]\( y \)[/tex]-value is [tex]\( 2 \)[/tex].
- For [tex]\( x = 2 \)[/tex], the corresponding [tex]\( y \)[/tex]-value is [tex]\( 2 \)[/tex].
Since the [tex]\( x \)[/tex]-value [tex]\( -2 \)[/tex] is associated with more than one [tex]\( y \)[/tex]-value, this relation does not represent a function.
4. Relation 4: [tex]\(\{(13,2),(13,3),(13,4),(13,5)\}\)[/tex]
- For [tex]\( x = 13 \)[/tex], there are four corresponding [tex]\( y \)[/tex]-values: [tex]\( 2 \)[/tex], [tex]\( 3 \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( 5 \)[/tex].
Since the [tex]\( x \)[/tex]-value [tex]\( 13 \)[/tex] is associated with more than one [tex]\( y \)[/tex]-value, this relation does not represent a function.
According to our analysis, none of the given relations represent a function. Therefore, there is no relation among the four provided that can be considered a function.
