Sure, let's analyze each set of pairs to determine whether they represent a function. As a reminder, a relation is considered a function if each [tex]\( x \)[/tex]-value in the relation is paired with exactly one [tex]\( y \)[/tex]-value.
1. First Table:
[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]
- Here, we have the [tex]\( x \)[/tex]-values: [tex]\(-5, -3, -3, 0, 5\)[/tex].
- Notice that the [tex]\( x \)[/tex]-value [tex]\(-3\)[/tex] is repeated, but paired with different [tex]\( y \)[/tex]-values [tex]\(5\)[/tex] and [tex]\(4\)[/tex].
- Since [tex]\(-3\)[/tex] is associated with two different [tex]\( y \)[/tex]-values, the first table does not represent a function.
2. Second Table:
[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]
- Here, we have the [tex]\( x \)[/tex]-values: [tex]\(-2, -1, 0, 0, 1\)[/tex].
- Notice that the [tex]\( x \)[/tex]-value [tex]\(0\)[/tex] is repeated, but paired with different [tex]\( y \)[/tex]-values [tex]\(-1\)[/tex] and [tex]\(0\)[/tex].
- Since [tex]\(0\)[/tex] is associated with two different [tex]\( y \)[/tex]-values, the second table does not represent a function.
3. Set of Pairs:
[tex]\[
\{(-12, 4), (-6, 10), (-4, 15), (-8, 18), (-12, 24)\}
\][/tex]
- Here, we have the [tex]\( x \)[/tex]-values: [tex]\(-12, -6, -4, -8, -12\)[/tex].
- Notice that the [tex]\( x \)[/tex]-value [tex]\(-12\)[/tex] is repeated, but paired with different [tex]\( y \)[/tex]-values [tex]\(4\)[/tex] and [tex]\(24\)[/tex].
- Since [tex]\(-12\)[/tex] is associated with two different [tex]\( y \)[/tex]-values, this set does not represent a function.
Therefore, none of the given tables or sets represent a function since each has at least one [tex]\( x \)[/tex]-value associated with more than one [tex]\( y \)[/tex]-value.
