To determine if the given relation defines [tex]\( y \)[/tex] as a one-to-one function of [tex]\( x \)[/tex], we need to check if each output value [tex]\( y \)[/tex] is associated with exactly one input value [tex]\( x \)[/tex]. In other words, [tex]\( y \)[/tex] must be unique for each [tex]\( x \)[/tex].
Given the relation:
[tex]\[ \{(-8, -3), (-13, 4), (-3, 0), (9, 3)\} \][/tex]
We can break this down into pairs of [tex]\( (x, y) \)[/tex] values:
1. [tex]\( (-8, -3) \)[/tex]
2. [tex]\( (-13, 4) \)[/tex]
3. [tex]\( (-3, 0) \)[/tex]
4. [tex]\( (9, 3) \)[/tex]
Let's take a look at the [tex]\( y \)[/tex]-values from these pairs:
1. [tex]\( -3 \)[/tex]
2. [tex]\( 4 \)[/tex]
3. [tex]\( 0 \)[/tex]
4. [tex]\( 3 \)[/tex]
Next, we check the uniqueness of these [tex]\( y \)[/tex]-values:
- [tex]\( -3 \)[/tex] appears once.
- [tex]\( 4 \)[/tex] appears once.
- [tex]\( 0 \)[/tex] appears once.
- [tex]\( 3 \)[/tex] appears once.
Since all [tex]\( y \)[/tex]-values are unique and no [tex]\( y \)[/tex]-value is repeated, each [tex]\( y \)[/tex] is paired with one, and only one, [tex]\( x \)[/tex].
Conclusively, the relation:
[tex]\[ \{(-8, -3), (-13, 4), (-3, 0), (9, 3)\} \][/tex]
does define [tex]\( y \)[/tex] as a one-to-one function of [tex]\( x \)[/tex].
