Answer :
To determine which of the given data sets/tables represent a function, we need to use the definition of a function. A relation is a function if and only if each input (or x-value) maps to exactly one output (or y-value). This means that no two pairs in the data can have the same x-value with different y-values.
Let's analyze each set/table step-by-step to see which ones meet this criterion.
### Table 1:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-5 & 10 \\ \hline-3 & 5 \\ \hline-3 & 4 \\ \hline 0 & 0 \\ \hline 5 & -10 \\ \hline \end{tabular} \][/tex]
In this table:
- The x-value -5 maps to 10.
- The x-value -3 maps to both 5 and 4.
Since -3 is associated with two different y-values (5 and 4), this table does not represent a function.
### Set 1:
[tex]\[ \{(-8,-2),(-4,1),(0,-2),(2,3),(4,-4)\} \][/tex]
In this set:
- Each x-value has a unique y-value:
- -8 maps to -2.
- -4 maps to 1.
- 0 maps to -2.
- 2 maps to 3.
- 4 maps to -4.
Since each x-value maps to exactly one y-value, this set represents a function.
### Table 2:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-2 & -3 \\ \hline-1 & -2 \\ \hline 0 & -1 \\ \hline 0 & 0 \\ \hline 1 & -1 \\ \hline \end{tabular} \][/tex]
In this table:
- The x-value 0 maps to both -1 and 0.
Since 0 is associated with two different y-values (-1 and 0), this table does not represent a function.
### Set 2:
[tex]\[ \begin{array}{c} \{(-12,4),(-6,10),(-4,15),(-8,18),(-12,24)\} \end{array} \][/tex]
In this set:
- The x-value -12 maps to both 4 and 24.
Since -12 is associated with two different y-values (4 and 24), this set does not represent a function.
### Summary:
- Table 1: Does not represent a function.
- Set 1: Represents a function.
- Table 2: Does not represent a function.
- Set 2: Does not represent a function.
Therefore, only the second data set (Set 1) represents a function.
Let's analyze each set/table step-by-step to see which ones meet this criterion.
### Table 1:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-5 & 10 \\ \hline-3 & 5 \\ \hline-3 & 4 \\ \hline 0 & 0 \\ \hline 5 & -10 \\ \hline \end{tabular} \][/tex]
In this table:
- The x-value -5 maps to 10.
- The x-value -3 maps to both 5 and 4.
Since -3 is associated with two different y-values (5 and 4), this table does not represent a function.
### Set 1:
[tex]\[ \{(-8,-2),(-4,1),(0,-2),(2,3),(4,-4)\} \][/tex]
In this set:
- Each x-value has a unique y-value:
- -8 maps to -2.
- -4 maps to 1.
- 0 maps to -2.
- 2 maps to 3.
- 4 maps to -4.
Since each x-value maps to exactly one y-value, this set represents a function.
### Table 2:
[tex]\[ \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline-2 & -3 \\ \hline-1 & -2 \\ \hline 0 & -1 \\ \hline 0 & 0 \\ \hline 1 & -1 \\ \hline \end{tabular} \][/tex]
In this table:
- The x-value 0 maps to both -1 and 0.
Since 0 is associated with two different y-values (-1 and 0), this table does not represent a function.
### Set 2:
[tex]\[ \begin{array}{c} \{(-12,4),(-6,10),(-4,15),(-8,18),(-12,24)\} \end{array} \][/tex]
In this set:
- The x-value -12 maps to both 4 and 24.
Since -12 is associated with two different y-values (4 and 24), this set does not represent a function.
### Summary:
- Table 1: Does not represent a function.
- Set 1: Represents a function.
- Table 2: Does not represent a function.
- Set 2: Does not represent a function.
Therefore, only the second data set (Set 1) represents a function.
