Let's determine which relations qualify as functions. A relation is considered a function if each input (or [tex]\( x \)[/tex]-value) corresponds to exactly one output (or [tex]\( y \)[/tex]-value). In other words, no [tex]\( x \)[/tex]-value repeats with different [tex]\( y \)[/tex]-values.
### Relation 1:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-10 & 10 \\
\hline
-5 & 5 \\
\hline
0 & 0 \\
\hline
5 & -5 \\
\hline
10 & -10 \\
\hline
\end{array}
\][/tex]
#### Analysis:
- [tex]\(-10 \mapsto 10\)[/tex]
- [tex]\(-5 \mapsto 5\)[/tex]
- [tex]\(0 \mapsto 0\)[/tex]
- [tex]\(5 \mapsto -5\)[/tex]
- [tex]\(10 \mapsto -10\)[/tex]
Each [tex]\( x \)[/tex]-value corresponds to a unique [tex]\( y \)[/tex]-value, so this is a function.
### Relation 2:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-3 & 2 \\
\hline
-1 & 1 \\
\hline
0 & 2 \\
\hline
0 & 1 \\
\hline
3 & 4 \\
\hline
\end{array}
\][/tex]
#### Analysis:
- [tex]\(-3 \mapsto 2\)[/tex]
- [tex]\(-1 \mapsto 1\)[/tex]
- [tex]\(0 \mapsto 2\)[/tex]
- [tex]\(0 \mapsto 1\)[/tex]
- [tex]\(3 \mapsto 4\)[/tex]
The [tex]\( x \)[/tex]-value [tex]\( 0 \)[/tex] corresponds to two different [tex]\( y \)[/tex]-values ([tex]\(2\)[/tex] and [tex]\(1\)[/tex]), so this is not a function.
### Relation 3:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-8 & -4 \\
\hline
-2 & -2 \\
\hline
1 & 3 \\
\hline
2 & 4 \\
\hline
4 & 6 \\
\hline
\end{array}
\][/tex]
#### Analysis:
- [tex]\(-8 \mapsto -4\)[/tex]
- [tex]\(-2 \mapsto -2\)[/tex]
- [tex]\(1 \mapsto 3\)[/tex]
- [tex]\(2 \mapsto 4\)[/tex]
- [tex]\(4 \mapsto 6\)[/tex]
Each [tex]\( x \)[/tex]-value corresponds to a unique [tex]\( y \)[/tex]-value, so this is a function.
### Relation 4:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 2 \\
\hline
1 & 4 \\
\hline
2 & 2 \\
\hline
2 & 6 \\
\hline
5 & 7 \\
\hline
\end{array}
\][/tex]
#### Analysis:
- [tex]\(0 \mapsto 2\)[/tex]
- [tex]\(1 \mapsto 4\)[/tex]
- [tex]\(2 \mapsto 2\)[/tex]
- [tex]\(2 \mapsto 6\)[/tex]
- [tex]\(5 \mapsto 7\)[/tex]
The [tex]\( x \)[/tex]-value [tex]\( 2 \)[/tex] corresponds to two different [tex]\( y \)[/tex]-values ([tex]\(2\)[/tex] and [tex]\(6\)[/tex]), so this is not a function.
### Conclusion:
- Relation 1 is a function.
- Relation 2 is not a function.
- Relation 3 is a function.
- Relation 4 is not a function.
So, the final determination is:
[tex]\[
(True, False, True, False)
\][/tex]
