To determine which table represents a function, we need to check if each input [tex]\(x\)[/tex] has exactly one corresponding output [tex]\(y\)[/tex]. A function assigns only one output to each input.
Let's evaluate each table one by one:
1. Table [tex]\(W\)
\[
\begin{array}{c|c}
x & y \\
\hline
-1 & 15 \\
-2 & 6 \\
0 & 15 \\
4 & 6 \\
\end{array}
\]
- \(x = -1 \rightarrow y = 15\)[/tex]
- [tex]\(x = -2 \rightarrow y = 6\)[/tex]
- [tex]\(x = 0 \rightarrow y = 15\)[/tex]
- [tex]\(x = 4 \rightarrow y = 6\)[/tex]
In this table, each [tex]\(x\)[/tex] has a unique [tex]\(y\)[/tex] value. This represents a function.
2. Table [tex]\(X\)
\[
\begin{array}{c|c}
x & y \\
\hline
0 & 4 \\
6 & 15 \\
0 & 6 \\
-2 & -1 \\
\end{array}
\]
- \(x = 0 \rightarrow y = 4\)[/tex]
- [tex]\(x = 6 \rightarrow y = 15\)[/tex]
- [tex]\(x = 0 \rightarrow y = 6\)[/tex]
- [tex]\(x = -2 \rightarrow y = -1\)[/tex]
In this table, [tex]\(x = 0\)[/tex] corresponds to two different [tex]\(y\)[/tex] values (4 and 6). This does not represent a function.
3. Table [tex]\(Y\)
\[
\begin{array}{c|c}
x & y \\
\hline
4 & 0 \\
-1 & -2 \\
6 & -2 \\
6 & 15 \\
\end{array}
\]
- \(x = 4 \rightarrow y = 0\)[/tex]
- [tex]\(x = -1 \rightarrow y = -2\)[/tex]
- [tex]\(x = 6 \rightarrow y = -2\)[/tex]
- [tex]\(x = 6 \rightarrow y = 15\)[/tex]
In this table, [tex]\(x = 6\)[/tex] corresponds to two different [tex]\(y\)[/tex] values (-2 and 15). This does not represent a function.
4. Table [tex]\(Z\)
\[
\begin{array}{c|c}
x & y \\
\hline
6 & -2 \\
4 & 0 \\
15 & -1 \\
4 & 3 \\
\end{array}
\]
- \(x = 6 \rightarrow y = -2\)[/tex]
- [tex]\(x = 4 \rightarrow y = 0\)[/tex]
- [tex]\(x = 15 \rightarrow y = -1\)[/tex]
- [tex]\(x = 4 \rightarrow y = 3\)[/tex]
In this table, [tex]\(x = 4\)[/tex] corresponds to two different [tex]\(y\)[/tex] values (0 and 3). This does not represent a function.
From the evaluation, only Table [tex]\(W\)[/tex] meets the requirement that each input [tex]\(x\)[/tex] has exactly one corresponding output [tex]\(y\)[/tex].
Therefore, the correct answer is:
A. [tex]\(W\)[/tex]
