Answer :
To determine which tables represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to understand that for [tex]\( y \)[/tex] to be a function of [tex]\( x \)[/tex], each value of [tex]\( x \)[/tex] must be associated with exactly one value of [tex]\( y \)[/tex]. Let's analyze each table:
1. First Table:
- [tex]\( x \)[/tex] values: 5, 7, 12
- [tex]\( y \)[/tex] values: 1, 6, 15
- Each value of [tex]\( x \)[/tex] (5, 7, and 12) is paired with just one value of [tex]\( y \)[/tex] (1, 6, and 15 respectively). There are no repeated [tex]\( x \)[/tex] values with different [tex]\( y \)[/tex] outcomes.
- Therefore, in the first table, [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex].
2. Second Table:
- [tex]\( x \)[/tex] values: 5, 7, 7
- [tex]\( y \)[/tex] values: 1, 6, 15
- Here, [tex]\( x = 7 \)[/tex] corresponds to two different [tex]\( y \)[/tex] values (6 and 15). This means that the same [tex]\( x \)[/tex] value does not have a unique [tex]\( y \)[/tex] value.
- Hence, in the second table, [tex]\( y \)[/tex] is not a function of [tex]\( x \)[/tex].
3. Third Table:
- [tex]\( x \)[/tex] values: 5, 7, 12
- [tex]\( y \)[/tex] values: 1, 6, 6
- Each value of [tex]\( x \)[/tex] (5, 7, and 12) is paired with one [tex]\( y \)[/tex] value (1, 6, and 6 respectively). Even though two different [tex]\( x \)[/tex] values (7 and 12) give the same [tex]\( y \)[/tex] value (6), this is acceptable because each [tex]\( x \)[/tex] has only one corresponding [tex]\( y \)[/tex].
- Therefore, in the third table, [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex].
In conclusion, the tables that represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] are the first and third tables.
1. First Table:
- [tex]\( x \)[/tex] values: 5, 7, 12
- [tex]\( y \)[/tex] values: 1, 6, 15
- Each value of [tex]\( x \)[/tex] (5, 7, and 12) is paired with just one value of [tex]\( y \)[/tex] (1, 6, and 15 respectively). There are no repeated [tex]\( x \)[/tex] values with different [tex]\( y \)[/tex] outcomes.
- Therefore, in the first table, [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex].
2. Second Table:
- [tex]\( x \)[/tex] values: 5, 7, 7
- [tex]\( y \)[/tex] values: 1, 6, 15
- Here, [tex]\( x = 7 \)[/tex] corresponds to two different [tex]\( y \)[/tex] values (6 and 15). This means that the same [tex]\( x \)[/tex] value does not have a unique [tex]\( y \)[/tex] value.
- Hence, in the second table, [tex]\( y \)[/tex] is not a function of [tex]\( x \)[/tex].
3. Third Table:
- [tex]\( x \)[/tex] values: 5, 7, 12
- [tex]\( y \)[/tex] values: 1, 6, 6
- Each value of [tex]\( x \)[/tex] (5, 7, and 12) is paired with one [tex]\( y \)[/tex] value (1, 6, and 6 respectively). Even though two different [tex]\( x \)[/tex] values (7 and 12) give the same [tex]\( y \)[/tex] value (6), this is acceptable because each [tex]\( x \)[/tex] has only one corresponding [tex]\( y \)[/tex].
- Therefore, in the third table, [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex].
In conclusion, the tables that represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] are the first and third tables.
