To determine which table represents a linear function, let's analyze the differences between consecutive [tex]\(y\)[/tex]-values for each table to see if they are constant.
1. First table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
y & 1 & 2 & 4 & 8 \\
\hline
\end{array}
\][/tex]
Differences:
[tex]\[
\begin{align*}
y_1 - y_0 & = 2 - 1 = 1 \\
y_2 - y_1 & = 4 - 2 = 2 \\
y_3 - y_2 & = 8 - 4 = 4 \\
\end{align*}
\][/tex]
The differences are not the same, so this is not a linear function.
2. Second table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
y & 4 & 3 & 1 & 2 \\
\hline
\end{array}
\][/tex]
Differences:
[tex]\[
\begin{align*}
y_1 - y_0 & = 3 - 4 = -1 \\
y_2 - y_1 & = 1 - 3 = -2 \\
y_3 - y_2 & = 2 - 1 = 1 \\
\end{align*}
\][/tex]
The differences are not the same, so this is not a linear function.
3. Third table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
y & 5 & 10 & 15 & 20 \\
\hline
\end{array}
\][/tex]
Differences:
[tex]\[
\begin{align*}
y_1 - y_0 & = 10 - 5 = 5 \\
y_2 - y_1 & = 15 - 10 = 5 \\
y_3 - y_2 & = 20 - 15 = 5 \\
\end{align*}
\][/tex]
The differences are all the same, so this is a linear function.
4. Fourth table:
[tex]\[
\begin{array}{|l|l|l|l|l|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
y & 0 & 1 & 0 & 3 \\
\hline
\end{array}
\][/tex]
Differences:
[tex]\[
\begin{align*}
y_1 - y_0 & = 1 - 0 = 1 \\
y_2 - y_1 & = 0 - 1 = -1 \\
y_3 - y_2 & = 3 - 0 = 3 \\
\end{align*}
\][/tex]
The differences are not the same, so this is not a linear function.
From our analysis, we found that only the third table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 \\
\hline
y & 5 & 10 & 15 & 20 \\
\hline
\end{array}
\][/tex]
represents a linear function, as the difference between consecutive [tex]\(y\)[/tex]-values is constant. Thus, the table that represents a linear function is the third one.
