Let's analyze the given relations to determine which one represents a function. A relation is defined as a function if each input maps to exactly one output.
Relation A:
Given in tabular form as:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 10 & 20 & 30 & 40 & 50 \\
\hline
y & 1 & 2 & 3 & 4 & 5 \\
\hline
\end{array}
\][/tex]
For Relation A, observe the pairs [tex]\((x,y)\)[/tex]:
[tex]\[
(10, 1), (20, 2), (30, 3), (40, 4), (50, 5)
\][/tex]
In this relation:
- Each [tex]\(x\)[/tex]-value (10, 20, 30, 40, 50) maps to exactly one unique [tex]\(y\)[/tex]-value (1, 2, 3, 4, 5) respectively.
- No [tex]\(x\)[/tex]-value is repeated.
- Therefore, every input [tex]\(x\)[/tex] is associated with exactly one output [tex]\(y\)[/tex].
Hence, Relation A represents a function.
Relation B:
Given as a set of pairs:
[tex]\[
\{(5,4),(4,3),(3,2),(2,1),(3,4)\}
\][/tex]
Observe the pairs:
[tex]\[
(5, 4), (4, 3), (3, 2), (2, 1), (3, 4)
\][/tex]
In this relation:
- The [tex]\(x\)[/tex]-value 3 maps to both 2 and 4.
- This means that the input 3 does not have a unique output as it corresponds to two different [tex]\(y\)[/tex]-values.
- Therefore, there is an [tex]\(x\)[/tex]-value (3) that is associated with more than one [tex]\(y\)[/tex]-value.
Hence, Relation B does not represent a function.
Summary:
After analyzing both relations, we find that only Relation A meets the criteria for being a function. Thus, the relation that represents a function is:
Relation A.
