Let's determine which of the given relations represents a function. A relation is considered a function if every [tex]\(x\)[/tex]-value has a unique [tex]\(y\)[/tex]-value associated with it, i.e., no [tex]\(x\)[/tex]-value is repeated with a different [tex]\(y\)[/tex]-value.
Consider the two relations given:
### Relation A
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 10 & 5 & 0 & 5 & 10 \\
\hline
$y$ & 1 & 2 & 3 & 4 & 5 \\
\hline
\end{tabular}
\][/tex]
In Relation A, we have the pairs: [tex]\((10, 1)\)[/tex], [tex]\((5, 2)\)[/tex], [tex]\((0, 3)\)[/tex], [tex]\((5, 4)\)[/tex], [tex]\((10, 5)\)[/tex].
- Here, the [tex]\(x\)[/tex]-value 10 appears twice, once with [tex]\(y\)[/tex]-value 1 and once with [tex]\(y\)[/tex]-value 5.
- Similarly, the [tex]\(x\)[/tex]-value 5 appears twice, once with [tex]\(y\)[/tex]-value 2 and once with [tex]\(y\)[/tex]-value 4.
Since the same [tex]\(x\)[/tex]-values are associated with different [tex]\(y\)[/tex]-values, Relation A does not satisfy the condition of a function. Therefore, Relation A is not a function.
### Relation B
[tex]\[
\{(6, 5), (5, 4), (4, 3), (3, 2), (5, 6)\}
\][/tex]
In Relation B, we have the pairs: [tex]\((6, 5)\)[/tex], [tex]\((5, 4)\)[/tex], [tex]\((4, 3)\)[/tex], [tex]\((3, 2)\)[/tex], [tex]\((5, 6)\)[/tex].
- Here, the [tex]\(x\)[/tex]-value 5 appears twice, once with [tex]\(y\)[/tex]-value 4 and once with [tex]\(y\)[/tex]-value 6.
Since the [tex]\(x\)[/tex]-value 5 is associated with two different [tex]\(y\)[/tex]-values, Relation B does not satisfy the condition of a function. Therefore, Relation B is not a function.
After evaluating both relations, we find that neither Relation A nor Relation B represents a function. Thus, the final conclusion is that neither of the given relations represents a function.
