At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To determine which tables represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] and are one-to-one, we need to confirm two things:
1. Each [tex]\( x \)[/tex] value must map to exactly one [tex]\( y \)[/tex] value.
2. Each [tex]\( y \)[/tex] value must be unique (no [tex]\( y \)[/tex] value is repeated for different [tex]\( x \)[/tex] values).
Let's analyze each table step-by-step:
### Table 1:
[tex]\[ \begin{array}{|r|r|r|r|} \hline x & 5 & 8 & 12 \\ \hline y & 3 & 10 & 15 \\ \hline \end{array} \][/tex]
- Condition 1: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex]: Each [tex]\( x \)[/tex] has a unique [tex]\( y \)[/tex].
- [tex]\( x = 5 \rightarrow y = 3 \)[/tex]
- [tex]\( x = 8 \rightarrow y = 10 \)[/tex]
- [tex]\( x = 12 \rightarrow y = 15 \)[/tex]
- Condition 2: One-to-one function: Each [tex]\( y \)[/tex] value is unique.
- [tex]\( y = 3 \)[/tex] for [tex]\( x = 5 \)[/tex]
- [tex]\( y = 10 \)[/tex] for [tex]\( x = 8 \)[/tex]
- [tex]\( y = 15 \)[/tex] for [tex]\( x = 12 \)[/tex]
Since both conditions are satisfied, Table 1 represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] and is one-to-one.
### Table 2:
[tex]\[ \begin{array}{|r|r|r|r|} \hline x & 5 & 8 & 12 \\ \hline y & 3 & 10 & 10 \\ \hline \end{array} \][/tex]
- Condition 1: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex]: Each [tex]\( x \)[/tex] has a unique [tex]\( y \)[/tex].
- [tex]\( x = 5 \rightarrow y = 3 \)[/tex]
- [tex]\( x = 8 \rightarrow y = 10 \)[/tex]
- [tex]\( x = 12 \rightarrow y = 10 \)[/tex]
- Condition 2: One-to-one function: Not satisfied because [tex]\( y \)[/tex] has repeated values.
- [tex]\( y = 3 \)[/tex] for [tex]\( x = 5 \)[/tex]
- [tex]\( y = 10 \)[/tex] for [tex]\( x = 8 \)[/tex]
- [tex]\( y = 10 \)[/tex] for [tex]\( x = 12 \)[/tex]
Since the second condition is not satisfied (repeated [tex]\( y \)[/tex] values), Table 2 does not represent [tex]\( y \)[/tex] as a one-to-one function of [tex]\( x \)[/tex].
### Table 3:
[tex]\[ \begin{array}{|r|r|r|r|} \hline x & 5 & 8 & 8 \\ \hline y & 3 & 10 & 15 \\ \hline \end{array} \][/tex]
- Condition 1: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex]: Not satisfied because [tex]\( x = 8 \)[/tex] maps to two different [tex]\( y \)[/tex] values.
- [tex]\( x = 5 \rightarrow y = 3 \)[/tex]
- [tex]\( x = 8 \rightarrow y = 10 \)[/tex]
- [tex]\( x = 8 \rightarrow y = 15 \)[/tex]
Thus, Table 3 does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], and it cannot be one-to-one.
### Conclusion
Only Table 1 represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] and is also one-to-one. Therefore, the correct answer is:
[tex]\[ \boxed{ \begin{array}{|r|r|r|} \hline x & 5 & 8 & 12 \\ \hline y & 3 & 10 & 15 \\ \hline \end{array} } \][/tex]
1. Each [tex]\( x \)[/tex] value must map to exactly one [tex]\( y \)[/tex] value.
2. Each [tex]\( y \)[/tex] value must be unique (no [tex]\( y \)[/tex] value is repeated for different [tex]\( x \)[/tex] values).
Let's analyze each table step-by-step:
### Table 1:
[tex]\[ \begin{array}{|r|r|r|r|} \hline x & 5 & 8 & 12 \\ \hline y & 3 & 10 & 15 \\ \hline \end{array} \][/tex]
- Condition 1: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex]: Each [tex]\( x \)[/tex] has a unique [tex]\( y \)[/tex].
- [tex]\( x = 5 \rightarrow y = 3 \)[/tex]
- [tex]\( x = 8 \rightarrow y = 10 \)[/tex]
- [tex]\( x = 12 \rightarrow y = 15 \)[/tex]
- Condition 2: One-to-one function: Each [tex]\( y \)[/tex] value is unique.
- [tex]\( y = 3 \)[/tex] for [tex]\( x = 5 \)[/tex]
- [tex]\( y = 10 \)[/tex] for [tex]\( x = 8 \)[/tex]
- [tex]\( y = 15 \)[/tex] for [tex]\( x = 12 \)[/tex]
Since both conditions are satisfied, Table 1 represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] and is one-to-one.
### Table 2:
[tex]\[ \begin{array}{|r|r|r|r|} \hline x & 5 & 8 & 12 \\ \hline y & 3 & 10 & 10 \\ \hline \end{array} \][/tex]
- Condition 1: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex]: Each [tex]\( x \)[/tex] has a unique [tex]\( y \)[/tex].
- [tex]\( x = 5 \rightarrow y = 3 \)[/tex]
- [tex]\( x = 8 \rightarrow y = 10 \)[/tex]
- [tex]\( x = 12 \rightarrow y = 10 \)[/tex]
- Condition 2: One-to-one function: Not satisfied because [tex]\( y \)[/tex] has repeated values.
- [tex]\( y = 3 \)[/tex] for [tex]\( x = 5 \)[/tex]
- [tex]\( y = 10 \)[/tex] for [tex]\( x = 8 \)[/tex]
- [tex]\( y = 10 \)[/tex] for [tex]\( x = 12 \)[/tex]
Since the second condition is not satisfied (repeated [tex]\( y \)[/tex] values), Table 2 does not represent [tex]\( y \)[/tex] as a one-to-one function of [tex]\( x \)[/tex].
### Table 3:
[tex]\[ \begin{array}{|r|r|r|r|} \hline x & 5 & 8 & 8 \\ \hline y & 3 & 10 & 15 \\ \hline \end{array} \][/tex]
- Condition 1: [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex]: Not satisfied because [tex]\( x = 8 \)[/tex] maps to two different [tex]\( y \)[/tex] values.
- [tex]\( x = 5 \rightarrow y = 3 \)[/tex]
- [tex]\( x = 8 \rightarrow y = 10 \)[/tex]
- [tex]\( x = 8 \rightarrow y = 15 \)[/tex]
Thus, Table 3 does not represent [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], and it cannot be one-to-one.
### Conclusion
Only Table 1 represents [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] and is also one-to-one. Therefore, the correct answer is:
[tex]\[ \boxed{ \begin{array}{|r|r|r|} \hline x & 5 & 8 & 12 \\ \hline y & 3 & 10 & 15 \\ \hline \end{array} } \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
