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]
