Answer :
To determine which set of ordered pairs represents a function, we need to recall the definition of a function in mathematical terms. A function from a set of inputs (domain) to a set of possible outputs (codomain) assigns each input exactly one output. In other words, each [tex]\( x \)[/tex]-value (input) should map to one and only one [tex]\( y \)[/tex]-value (output).
Let's analyze each set to see if any of the [tex]\( x \)[/tex]-values are repeated with different [tex]\( y \)[/tex]-values.
### First Set: [tex]\(\{(2, -2), (1, 5), (-2, 2), (1, -3), (8, -1)\}\)[/tex]
- [tex]\( (2, -2) \)[/tex]: [tex]\( x = 2 \)[/tex]
- [tex]\( (1, 5) \)[/tex]: [tex]\( x = 1 \)[/tex]
- [tex]\( (-2, 2) \)[/tex]: [tex]\( x = -2 \)[/tex]
- [tex]\( (1, -3) \)[/tex]: [tex]\( x = 1 \)[/tex] (repeated [tex]\( x \)[/tex]-value with different [tex]\( y \)[/tex]-values, [tex]\( y = 5 \)[/tex] and [tex]\( y = -3 \)[/tex])
- [tex]\( (8, -1) \)[/tex]: [tex]\( x = 8 \)[/tex]
Since [tex]\( x = 1 \)[/tex] is repeated with different [tex]\( y \)[/tex]-values ([tex]\( 5 \)[/tex] and [tex]\( -3 \)[/tex]), the first set does not represent a function.
### Second Set: [tex]\(\{(3, -1), (7, 1), (-6, -1), (9, 1), (2, -1)\}\)[/tex]
- [tex]\( (3, -1) \)[/tex]: [tex]\( x = 3 \)[/tex]
- [tex]\( (7, 1) \)[/tex]: [tex]\( x = 7 \)[/tex]
- [tex]\( (-6, -1) \)[/tex]: [tex]\( x = -6 \)[/tex]
- [tex]\( (9, 1) \)[/tex]: [tex]\( x = 9 \)[/tex]
- [tex]\( (2, -1) \)[/tex]: [tex]\( x = 2 \)[/tex]
All [tex]\( x \)[/tex]-values are unique, meaning each [tex]\( x \)[/tex]-value maps to exactly one [tex]\( y \)[/tex]-value. Therefore, the second set represents a function.
### Third Set: [tex]\(\{(6, 8), (5, 2), (-2, -5), (1, -3), (-2, 9)\}\)[/tex]
- [tex]\( (6, 8) \)[/tex]: [tex]\( x = 6 \)[/tex]
- [tex]\( (5, 2) \)[/tex]: [tex]\( x = 5 \)[/tex]
- [tex]\( (-2, -5) \)[/tex]: [tex]\( x = -2 \)[/tex]
- [tex]\( (1, -3) \)[/tex]: [tex]\( x = 1 \)[/tex]
- [tex]\( (-2, 9) \)[/tex]: [tex]\( x = -2 \)[/tex] (repeated [tex]\( x \)[/tex]-value with different [tex]\( y \)[/tex]-values, [tex]\( y = -5 \)[/tex] and [tex]\( y = 9 \)[/tex])
Since [tex]\( x = -2 \)[/tex] is repeated with different [tex]\( y \)[/tex]-values ([tex]\( -5 \)[/tex] and [tex]\( 9 \)[/tex]), the third set does not represent a function.
### Fourth Set: [tex]\(\{(-3, 1), (6, 3), (-3, 2), (-3, -3), (1, -1)\}\)[/tex]
- [tex]\( (-3, 1) \)[/tex]: [tex]\( x = -3 \)[/tex]
- [tex]\( (6, 3) \)[/tex]: [tex]\( x = 6 \)[/tex]
- [tex]\( (-3, 2) \)[/tex]: [tex]\( x = -3 \)[/tex] (repeated [tex]\( x \)[/tex]-value with different [tex]\( y \)[/tex]-values, [tex]\( y = 1 \)[/tex], [tex]\( y = 2 \)[/tex], and [tex]\( y = -3 \)[/tex])
- [tex]\( (-3, -3) \)[/tex]: [tex]\( x = -3 \)[/tex] (repeated again)
- [tex]\( (1, -1) \)[/tex]: [tex]\( x = 1 \)[/tex]
Since [tex]\( x = -3 \)[/tex] is repeated multiple times with different [tex]\( y \)[/tex]-values ([tex]\( 1 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( -3 \)[/tex]), the fourth set does not represent a function.
### Conclusion
After analyzing all four sets, only the second set [tex]\(\{(3, -1), (7, 1), (-6, -1), (9, 1), (2, -1)\}\)[/tex] represents a function, as no [tex]\( x \)[/tex]-value is repeated with different [tex]\( y \)[/tex]-values.
Let's analyze each set to see if any of the [tex]\( x \)[/tex]-values are repeated with different [tex]\( y \)[/tex]-values.
### First Set: [tex]\(\{(2, -2), (1, 5), (-2, 2), (1, -3), (8, -1)\}\)[/tex]
- [tex]\( (2, -2) \)[/tex]: [tex]\( x = 2 \)[/tex]
- [tex]\( (1, 5) \)[/tex]: [tex]\( x = 1 \)[/tex]
- [tex]\( (-2, 2) \)[/tex]: [tex]\( x = -2 \)[/tex]
- [tex]\( (1, -3) \)[/tex]: [tex]\( x = 1 \)[/tex] (repeated [tex]\( x \)[/tex]-value with different [tex]\( y \)[/tex]-values, [tex]\( y = 5 \)[/tex] and [tex]\( y = -3 \)[/tex])
- [tex]\( (8, -1) \)[/tex]: [tex]\( x = 8 \)[/tex]
Since [tex]\( x = 1 \)[/tex] is repeated with different [tex]\( y \)[/tex]-values ([tex]\( 5 \)[/tex] and [tex]\( -3 \)[/tex]), the first set does not represent a function.
### Second Set: [tex]\(\{(3, -1), (7, 1), (-6, -1), (9, 1), (2, -1)\}\)[/tex]
- [tex]\( (3, -1) \)[/tex]: [tex]\( x = 3 \)[/tex]
- [tex]\( (7, 1) \)[/tex]: [tex]\( x = 7 \)[/tex]
- [tex]\( (-6, -1) \)[/tex]: [tex]\( x = -6 \)[/tex]
- [tex]\( (9, 1) \)[/tex]: [tex]\( x = 9 \)[/tex]
- [tex]\( (2, -1) \)[/tex]: [tex]\( x = 2 \)[/tex]
All [tex]\( x \)[/tex]-values are unique, meaning each [tex]\( x \)[/tex]-value maps to exactly one [tex]\( y \)[/tex]-value. Therefore, the second set represents a function.
### Third Set: [tex]\(\{(6, 8), (5, 2), (-2, -5), (1, -3), (-2, 9)\}\)[/tex]
- [tex]\( (6, 8) \)[/tex]: [tex]\( x = 6 \)[/tex]
- [tex]\( (5, 2) \)[/tex]: [tex]\( x = 5 \)[/tex]
- [tex]\( (-2, -5) \)[/tex]: [tex]\( x = -2 \)[/tex]
- [tex]\( (1, -3) \)[/tex]: [tex]\( x = 1 \)[/tex]
- [tex]\( (-2, 9) \)[/tex]: [tex]\( x = -2 \)[/tex] (repeated [tex]\( x \)[/tex]-value with different [tex]\( y \)[/tex]-values, [tex]\( y = -5 \)[/tex] and [tex]\( y = 9 \)[/tex])
Since [tex]\( x = -2 \)[/tex] is repeated with different [tex]\( y \)[/tex]-values ([tex]\( -5 \)[/tex] and [tex]\( 9 \)[/tex]), the third set does not represent a function.
### Fourth Set: [tex]\(\{(-3, 1), (6, 3), (-3, 2), (-3, -3), (1, -1)\}\)[/tex]
- [tex]\( (-3, 1) \)[/tex]: [tex]\( x = -3 \)[/tex]
- [tex]\( (6, 3) \)[/tex]: [tex]\( x = 6 \)[/tex]
- [tex]\( (-3, 2) \)[/tex]: [tex]\( x = -3 \)[/tex] (repeated [tex]\( x \)[/tex]-value with different [tex]\( y \)[/tex]-values, [tex]\( y = 1 \)[/tex], [tex]\( y = 2 \)[/tex], and [tex]\( y = -3 \)[/tex])
- [tex]\( (-3, -3) \)[/tex]: [tex]\( x = -3 \)[/tex] (repeated again)
- [tex]\( (1, -1) \)[/tex]: [tex]\( x = 1 \)[/tex]
Since [tex]\( x = -3 \)[/tex] is repeated multiple times with different [tex]\( y \)[/tex]-values ([tex]\( 1 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( -3 \)[/tex]), the fourth set does not represent a function.
### Conclusion
After analyzing all four sets, only the second set [tex]\(\{(3, -1), (7, 1), (-6, -1), (9, 1), (2, -1)\}\)[/tex] represents a function, as no [tex]\( x \)[/tex]-value is repeated with different [tex]\( y \)[/tex]-values.
