To determine the type of function represented by the given table of values, we need to analyze the relationship between the [tex]\(x\)[/tex]-values and the corresponding [tex]\(y\)[/tex]-values.
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 4 \\
\hline
2 & 8 \\
\hline
3 & 12 \\
\hline
4 & 16 \\
\hline
5 & 20 \\
\hline
\end{array}
\][/tex]
Here's a step-by-step process to determine the nature of the function:
1. Calculate the Differences in [tex]\(x\)[/tex]-Values:
We compute the differences between consecutive [tex]\(x\)[/tex]-values.
[tex]\[
\begin{aligned}
x_2 - x_1 &= 2 - 1 = 1 \\
x_3 - x_2 &= 3 - 2 = 1 \\
x_4 - x_3 &= 4 - 3 = 1 \\
x_5 - x_4 &= 5 - 4 = 1 \\
\end{aligned}
\][/tex]
These differences are all equal to 1.
2. Calculate the Differences in [tex]\(y\)[/tex]-Values:
Next, we compute the differences between consecutive [tex]\(y\)[/tex]-values.
[tex]\[
\begin{aligned}
y_2 - y_1 &= 8 - 4 = 4 \\
y_3 - y_2 &= 12 - 8 = 4 \\
y_4 - y_3 &= 16 - 12 = 4 \\
y_5 - y_4 &= 20 - 16 = 4 \\
\end{aligned}
\][/tex]
These differences are all equal to 4.
3. Analyze the Differences:
In a linear function, the difference between consecutive [tex]\(y\)[/tex]-values (also known as the first differences) should be constant when the [tex]\(x\)[/tex]-values are evenly spaced.
For the given table, the differences between consecutive [tex]\(x\)[/tex]-values are constant ([tex]\(1\)[/tex]), and the differences between consecutive [tex]\(y\)[/tex]-values are also constant ([tex]\(4\)[/tex]).
This consistency in the differences signifies a linear relationship.
4. Conclusion:
Since the differences between the [tex]\(y\)[/tex]-values are constant, the function represented by the table is linear.
Thus, the type of function is linear. The correct answer is:
D. linear
