Let's evaluate each given relation to determine if they represent functions.
### Explanation:
A relation is a function if for every [tex]\( x \)[/tex] value there is only one corresponding [tex]\( y \)[/tex] value. In other words, no [tex]\( x \)[/tex] value is paired with more than one [tex]\( y \)[/tex] value.
### Solutions:
1. Relation:
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $y$ \\
\hline 3 & 9 \\
\hline 8 & 7 \\
\hline 12 & 5 \\
\hline 15 & 3 \\
\hline
\end{tabular}
\][/tex]
Is this a function? Yes
Why? Each [tex]\( x \)[/tex] value in the relation (3, 8, 12, 15) maps to exactly one [tex]\( y \)[/tex] value. They are all unique and do not repeat.
2. Relation:
[tex]\[
\{(8,5),(6,6),(4,7),(2,8)\}
\][/tex]
Is this a function? Yes
Why? Each [tex]\( x \)[/tex] value in the set (8, 6, 4, 2) maps to exactly one [tex]\( y \)[/tex] value. There are no duplicate [tex]\( x \)[/tex] values with different [tex]\( y \)[/tex] values.
3. Relation:
[tex]\[
\{(8,5),(7,7),(6,9),(5,11)\}
\][/tex]
Is this a function? Yes
Why? Each [tex]\( x \)[/tex] value in the set (8, 7, 6, 5) maps to exactly one [tex]\( y \)[/tex] value. There are no duplicate [tex]\( x \)[/tex] values with different [tex]\( y \)[/tex] values.
4. Relation:
[tex]\[
\{(1,7),(0,5),(1,3),(2,1)\}
\][/tex]
Is this a function? No
Why? The [tex]\( x \)[/tex] value 1 is associated with both 7 and 3, violating the rule that each [tex]\( x \)[/tex] value must map to exactly one [tex]\( y \)[/tex] value.
5. Relation:
[tex]\[
\begin{tabular}{|c|c|}
\hline$x$ & $y$ \\
\hline-1 & 6 \\
\hline 0 & 5 \\
\hline 1 & 6 \\
\hline 2 & 5 \\
\hline
\end{tabular}
\][/tex]
Is this a function? Yes
Why? Each [tex]\( x \)[/tex] value in the table (-1, 0, 1, 2) maps to exactly one [tex]\( y \)[/tex] value. They are unique and do not repeat.
6. Relation:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline$x$ & 1 & 2 & 3 & 2 & 1 \\
\hline$y$ & 12 & 8 & 5 & 3 & 2 \\
\hline
\end{tabular}
\][/tex]
Is this a function? No
Why? The [tex]\( x \)[/tex] values 1 and 2 each appear more than once with different [tex]\( y \)[/tex] values (1 maps to both 12 and 2, and 2 maps to both 8 and 3), which means that it does not satisfy the condition of having each [tex]\( x \)[/tex] value map to exactly one [tex]\( y \)[/tex] value.
### Summary:
1. Relation 1: Yes
2. Relation 8: Yes
3. Relation 7: Yes
4. Relation 10: No
5. Relation 11: Yes
6. Relation 6: No
