To determine which of the given sets of ordered pairs represents a function, we must understand that a relation is a function if and only if each input (or first element of each pair, which is the [tex]\( x \)[/tex]-value) is associated with exactly one output (or second element of each pair, which is the [tex]\( y \)[/tex]-value). In other words, there should be no repeated [tex]\( x \)[/tex]-values with different [tex]\( y \)[/tex]-values.
Let's analyze each set of pairs individually to check if they are functions.
1. Set 1: [tex]\(\{(5, 4), (5, 6), (5, 8), (5, 10), (5, 12)\}\)[/tex]
- Here, all pairs have [tex]\( x = 5 \)[/tex].
- The [tex]\( x \)[/tex]-value 5 is associated with different [tex]\( y \)[/tex]-values (4, 6, 8, 10, and 12).
- This means that an input of 5 corresponds to multiple outputs, which violates the definition of a function.
- Conclusion: This set is not a function.
2. Set 2: [tex]\(\{(-3, -2), (-2, -1), (0, -1), (0, 1), (1, 2)\}\)[/tex]
- Here, the [tex]\( x \)[/tex]-value 0 is repeated with [tex]\( y = -1 \)[/tex] and [tex]\( y = 1 \)[/tex].
- This means that an input of 0 corresponds to different outputs, which again violates the definition of a function.
- Conclusion: This set is not a function.
3. Set 3: [tex]\(\{(5, 2), (-4, 2), (3, 6), (0, 4), (-1, 2)\}\)[/tex]
- In this set, each [tex]\( x \)[/tex]-value (5, -4, 3, 0, -1) is paired uniquely with a [tex]\( y \)[/tex]-value.
- There are no repeated [tex]\( x \)[/tex]-values.
- Each input is associated with exactly one output.
- Conclusion: This set is a function.
4. Set 4: [tex]\(\{(7, 3), (-6, 8), (-3, 5), (0, -3), (7, 11)\}\)[/tex]
- Here, the [tex]\( x \)[/tex]-value 7 is repeated with [tex]\( y = 3 \)[/tex] and [tex]\( y = 11 \)[/tex].
- This means that an input of 7 corresponds to different outputs, which violates the definition of a function.
- Conclusion: This set is not a function.
Summarizing the analysis:
- [tex]\(\{(5, 4), (5, 6), (5, 8), (5, 10), (5, 12)\}\)[/tex] is not a function.
- [tex]\(\{(-3, -2), (-2, -1), (0, -1), (0, 1), (1, 2)\}\)[/tex] is not a function.
- [tex]\(\{(5, 2), (-4, 2), (3, 6), (0, 4), (-1, 2)\}\)[/tex] is a function.
- [tex]\(\{(7, 3), (-6, 8), (-3, 5), (0, -3), (7, 11)\}\)[/tex] is not a function.
Hence, the only relation that is a function is the third set:
[tex]\[
\{(5,2),(-4,2),(3,6),(0,4),(-1,2)\}
\][/tex]
