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Sagot :
To determine which of these sets of ordered pairs represent a function, we need to verify that each [tex]\( x \)[/tex] value (input) maps to exactly one [tex]\( y \)[/tex] value (output). This means for a set of pairs to represent a function, no [tex]\( x \)[/tex] value should be repeated with different [tex]\( y \)[/tex] values.
Let's analyze each set of ordered pairs step by step.
### Set 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -5 & 10 \\ \hline -3 & 5 \\ \hline -3 & 4 \\ \hline 0 & 0 \\ \hline 5 & -10 \\ \hline \end{array} \][/tex]
In this set:
- [tex]\( x = -5 \)[/tex] maps to [tex]\( y = 10 \)[/tex]
- [tex]\( x = -3 \)[/tex] maps to [tex]\( y = 5 \)[/tex]
- [tex]\( x = -3 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
- [tex]\( x = 5 \)[/tex] maps to [tex]\( y = -10 \)[/tex]
Here, [tex]\( x = -3 \)[/tex] is repeated with different [tex]\( y \)[/tex] values (5 and 4). Therefore, this set does not represent a function.
### Set 2
[tex]\[ \{(-8,-2),(-4,1),(0,-2),(2,3),(4,-4)\} \][/tex]
In this set:
- [tex]\( x = -8 \)[/tex] maps to [tex]\( y = -2 \)[/tex]
- [tex]\( x = -4 \)[/tex] maps to [tex]\( y = 1 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = -2 \)[/tex]
- [tex]\( x = 2 \)[/tex] maps to [tex]\( y = 3 \)[/tex]
- [tex]\( x = 4 \)[/tex] maps to [tex]\( y = -4 \)[/tex]
Each [tex]\( x \)[/tex] value is unique and maps to one [tex]\( y \)[/tex] value. Therefore, this set does represent a function.
### Set 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & -3 \\ \hline -1 & -2 \\ \hline 0 & -1 \\ \hline 0 & 0 \\ \hline 1 & -1 \\ \hline \end{array} \][/tex]
In this set:
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = -3 \)[/tex]
- [tex]\( x = -1 \)[/tex] maps to [tex]\( y = -2 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = -1 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
- [tex]\( x = 1 \)[/tex] maps to [tex]\( y = -1 \)[/tex]
Here, [tex]\( x = 0 \)[/tex] is repeated with different [tex]\( y \)[/tex] values (-1 and 0). Therefore, this set does not represent a function.
### Set 4
[tex]\[ \{(-12,4),(-6,10),(-4,15),(-8,18),(-12,24)\} \][/tex]
In this set:
- [tex]\( x = -12 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = -6 \)[/tex] maps to [tex]\( y = 10 \)[/tex]
- [tex]\( x = -4 \)[/tex] maps to [tex]\( y = 15 \)[/tex]
- [tex]\( x = -8 \)[/tex] maps to [tex]\( y = 18 \)[/tex]
- [tex]\( x = -12 \)[/tex] maps to [tex]\( y = 24 \)[/tex]
Here, [tex]\( x = -12 \)[/tex] is repeated with different [tex]\( y \)[/tex] values (4 and 24). Therefore, this set does not represent a function.
### Conclusion
Only the second set of points represents a function, as each [tex]\( x \)[/tex] value maps to exactly one [tex]\( y \)[/tex] value:
- Set 1: Not a function
- Set 2: Function
- Set 3: Not a function
- Set 4: Not a function
Therefore, the results are:
[tex]\[ (0, 1, 0, 0) \][/tex]
Let's analyze each set of ordered pairs step by step.
### Set 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -5 & 10 \\ \hline -3 & 5 \\ \hline -3 & 4 \\ \hline 0 & 0 \\ \hline 5 & -10 \\ \hline \end{array} \][/tex]
In this set:
- [tex]\( x = -5 \)[/tex] maps to [tex]\( y = 10 \)[/tex]
- [tex]\( x = -3 \)[/tex] maps to [tex]\( y = 5 \)[/tex]
- [tex]\( x = -3 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
- [tex]\( x = 5 \)[/tex] maps to [tex]\( y = -10 \)[/tex]
Here, [tex]\( x = -3 \)[/tex] is repeated with different [tex]\( y \)[/tex] values (5 and 4). Therefore, this set does not represent a function.
### Set 2
[tex]\[ \{(-8,-2),(-4,1),(0,-2),(2,3),(4,-4)\} \][/tex]
In this set:
- [tex]\( x = -8 \)[/tex] maps to [tex]\( y = -2 \)[/tex]
- [tex]\( x = -4 \)[/tex] maps to [tex]\( y = 1 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = -2 \)[/tex]
- [tex]\( x = 2 \)[/tex] maps to [tex]\( y = 3 \)[/tex]
- [tex]\( x = 4 \)[/tex] maps to [tex]\( y = -4 \)[/tex]
Each [tex]\( x \)[/tex] value is unique and maps to one [tex]\( y \)[/tex] value. Therefore, this set does represent a function.
### Set 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -2 & -3 \\ \hline -1 & -2 \\ \hline 0 & -1 \\ \hline 0 & 0 \\ \hline 1 & -1 \\ \hline \end{array} \][/tex]
In this set:
- [tex]\( x = -2 \)[/tex] maps to [tex]\( y = -3 \)[/tex]
- [tex]\( x = -1 \)[/tex] maps to [tex]\( y = -2 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = -1 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = 0 \)[/tex]
- [tex]\( x = 1 \)[/tex] maps to [tex]\( y = -1 \)[/tex]
Here, [tex]\( x = 0 \)[/tex] is repeated with different [tex]\( y \)[/tex] values (-1 and 0). Therefore, this set does not represent a function.
### Set 4
[tex]\[ \{(-12,4),(-6,10),(-4,15),(-8,18),(-12,24)\} \][/tex]
In this set:
- [tex]\( x = -12 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = -6 \)[/tex] maps to [tex]\( y = 10 \)[/tex]
- [tex]\( x = -4 \)[/tex] maps to [tex]\( y = 15 \)[/tex]
- [tex]\( x = -8 \)[/tex] maps to [tex]\( y = 18 \)[/tex]
- [tex]\( x = -12 \)[/tex] maps to [tex]\( y = 24 \)[/tex]
Here, [tex]\( x = -12 \)[/tex] is repeated with different [tex]\( y \)[/tex] values (4 and 24). Therefore, this set does not represent a function.
### Conclusion
Only the second set of points represents a function, as each [tex]\( x \)[/tex] value maps to exactly one [tex]\( y \)[/tex] value:
- Set 1: Not a function
- Set 2: Function
- Set 3: Not a function
- Set 4: Not a function
Therefore, the results are:
[tex]\[ (0, 1, 0, 0) \][/tex]
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