To determine whether each given relation represents a function, we need to check if every input value (x-coordinate) maps to exactly one output value (y-coordinate).
### Relation 1 — Represented by the table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -10 & -9 & 0 & 5 \\
\hline
y & 19 & 17 & 25 & 19 \\
\hline
\end{array}
\][/tex]
In this table:
- [tex]\( x = -10 \)[/tex] maps to [tex]\( y = 19 \)[/tex]
- [tex]\( x = -9 \)[/tex] maps to [tex]\( y = 17 \)[/tex]
- [tex]\( x = 0 \)[/tex] maps to [tex]\( y = 25 \)[/tex]
- [tex]\( x = 5 \)[/tex] maps to [tex]\( y = 19 \)[/tex]
We observe that each input [tex]\( x \)[/tex] value is unique and maps to exactly one [tex]\( y \)[/tex] value. Therefore, this relation does represent a function.
### Relation 2 — Represented by the list of tuples:
[tex]\[ ((0,0),(-4,3),(7,1),(-4,5),(3,2)) \][/tex]
Here, we have the set of pairs:
- [tex]\((0, 0)\)[/tex]
- [tex]\((-4, 3)\)[/tex]
- [tex]\((7, 1)\)[/tex]
- [tex]\((-4, 5)\)[/tex]
- [tex]\((3, 2)\)[/tex]
We need to check each input [tex]\( x \)[/tex] value:
- [tex]\(0\)[/tex] maps to [tex]\(0\)[/tex]
- [tex]\(-4\)[/tex] maps to [tex]\(3\)[/tex] and [tex]\(-4\)[/tex] also maps to [tex]\(5\)[/tex]
The input value [tex]\(-4\)[/tex] maps to two different outputs ([tex]\(3\)[/tex] and [tex]\(5\)[/tex]). This means that for the same input [tex]\(x = -4\)[/tex], there are multiple outputs.
Hence, this relation does not represent a function because a single input maps to more than one output.
### Conclusion:
- The table [tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -10 & -9 & 0 & 5 \\
\hline
y & 19 & 17 & 25 & 19 \\
\hline
\end{array}
\][/tex] represents a function.
- The list of tuples [tex]\(((0, 0), (-4, 3), (7, 1), (-4, 5), (3, 2))\)[/tex] does not represent a function.
