Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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]
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]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
