To determine whether the given table represents a function, we need to understand the definition of a function in mathematics. A function from set [tex]\( X \)[/tex] to set [tex]\( Y \)[/tex] allocates each element in [tex]\( X \)[/tex] to exactly one element in [tex]\( Y \)[/tex]. Essentially, an [tex]\( x \)[/tex]-value should have exactly one corresponding [tex]\( y \)[/tex]-value.
Let's examine the given table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 1 \\
\hline
2 & 4 \\
\hline
3 & 4 \\
\hline
4 & 2 \\
\hline
5 & 5 \\
\hline
\end{array}
\][/tex]
Look at the [tex]\( x \)[/tex]-values and their corresponding [tex]\( y \)[/tex]-values:
- [tex]\( x = 2 \)[/tex] maps to [tex]\( y = 1 \)[/tex]
- [tex]\( x = 2 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = 3 \)[/tex] maps to [tex]\( y = 4 \)[/tex]
- [tex]\( x = 4 \)[/tex] maps to [tex]\( y = 2 \)[/tex]
- [tex]\( x = 5 \)[/tex] maps to [tex]\( y = 5 \)[/tex]
We can observe that the [tex]\( x \)[/tex]-value 2 maps to two different [tex]\( y \)[/tex]-values: 1 and 4.
According to the definition of a function, this violates the requirement that each [tex]\( x \)[/tex]-value must correspond to exactly one [tex]\( y \)[/tex]-value. Because there is at least one [tex]\( x \)[/tex]-value that maps to more than one [tex]\( y \)[/tex]-value, the given table does not represent a function.
Therefore, the correct answer is:
A. No, because one [tex]\( x \)[/tex]-value corresponds to two different [tex]\( y \)[/tex]-values.
