To determine whether a relation is a function, we need to verify that each input [tex]\( x \)[/tex]-value maps to exactly one output [tex]\( y \)[/tex]-value. Let's analyze the given relation:
[tex]\[
\{ (0,0), (1,1), (1,2), (2,3), (2,4), (3,5) \}
\][/tex]
Step-by-Step Solution:
1. Identify the inputs (x-values):
- Here the x-values are [tex]\( 0, 1, 1, 2, 2, 3 \)[/tex].
2. Check for duplicates among the x-values:
- The input [tex]\( 1 \)[/tex] appears twice, mapping to both [tex]\( 1 \)[/tex] and [tex]\( 2 \)[/tex].
- The input [tex]\( 2 \)[/tex] also appears twice, mapping to both [tex]\( 3 \)[/tex] and [tex]\( 4 \)[/tex].
3. Determine if each x-value maps uniquely to one y-value:
- Since there are duplicates among the x-values (specifically [tex]\( 1 \)[/tex] and [tex]\( 2 \)[/tex]), and they each map to different y-values (i.e., [tex]\( 1 \rightarrow 1 \)[/tex] and [tex]\( 1 \rightarrow 2 \)[/tex], [tex]\( 2 \rightarrow 3 \)[/tex] and [tex]\( 2 \rightarrow 4 \)[/tex]), this relation is not a function.
Next, let's examine the tabulated relation:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-1 & 4 \\
\hline
-2 & 3 \\
\hline
-3 & 2 \\
\hline
\end{array}
\][/tex]
Step-by-Step Solution:
1. Identify the inputs (x-values):
- The x-values here are [tex]\(-1\)[/tex], [tex]\(-2\)[/tex], and [tex]\(-3\)[/tex].
2. Check for duplicates among the x-values:
- There are no duplicates; each x-value appears exactly once.
3. Determine if each x-value maps uniquely to one y-value:
- Each x-value maps to exactly one y-value:
- [tex]\(-1 \rightarrow 4\)[/tex]
- [tex]\(-2 \rightarrow 3\)[/tex]
- [tex]\(-3 \rightarrow 2\)[/tex]
Since each x-value maps to exactly one y-value, this relation is indeed a function.
Conclusion:
The relation represented by the table is a function, whereas the set [tex]\(\{ (0,0), (1,1), (1,2), (2,3), (2,4), (3,5) \}\)[/tex] is not a function.
