To determine whether a given relation is a function, we need to check if each input [tex]\( x \)[/tex] has exactly one output [tex]\( y \)[/tex]. In other words, for a relation to be a function, there should be no repeated [tex]\( x \)[/tex]-values paired with different [tex]\( y \)[/tex]-values.
Let's examine the given relation:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 4 \\
\hline
-1 & -2 \\
\hline
3 & 10 \\
\hline
5 & 16 \\
\hline
\end{array}
\][/tex]
1. First, list all the [tex]\( x \)[/tex]-values: [tex]\( 1, -1, 3, 5 \)[/tex].
2. Check for any repeated [tex]\( x \)[/tex]-values:
- [tex]\( 1 \)[/tex] appears once.
- [tex]\(-1 \)[/tex] appears once.
- [tex]\( 3 \)[/tex] appears once.
- [tex]\( 5 \)[/tex] appears once.
Since all the [tex]\( x \)[/tex]-values are unique and there are no repetitions, each [tex]\( x \)[/tex]-value is associated with exactly one [tex]\( y \)[/tex]-value.
Therefore, the relation given in the table satisfies the condition of a function, where each input [tex]\( x \)[/tex] has one and only one output [tex]\( y \)[/tex].
Hence, the given relation is a function.
Answer: Yes
