To determine which function increases at the fastest rate between [tex]\(x=0\)[/tex] and [tex]\(x=8\)[/tex], we'll analyze the differences between consecutive values for both functions.
### Linear Function [tex]\(f(x) = 2x + 2\)[/tex]
We evaluate the linear function at points from 0 to 8:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 2 \\
1 & 4 \\
2 & 6 \\
3 & 8 \\
4 & 10 \\
5 & 12 \\
6 & 14 \\
7 & 16 \\
8 & 18 \\
\hline
\end{array}
\][/tex]
Next, we calculate the differences between consecutive [tex]\(f(x)\)[/tex] values:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Interval} & \text{Difference} \\
\hline
1-0 & f(1) - f(0) = 4 - 2 = 2 \\
2-1 & f(2) - f(1) = 6 - 4 = 2 \\
3-2 & f(3) - f(2) = 8 - 6 = 2 \\
4-3 & f(4) - f(3) = 10 - 8 = 2 \\
5-4 & f(5) - f(4) = 12 - 10 = 2 \\
6-5 & f(6) - f(5) = 14 - 12 = 2 \\
7-6 & f(7) - f(6) = 16 - 14 = 2 \\
8-7 & f(8) - f(7) = 18 - 16 = 2 \\
\hline
\end{array}
\][/tex]
All differences for the linear function are equal to 2. Therefore, the maximum difference for the linear function is:
[tex]\[
\max_{\text{linear}} = 2
\][/tex]
### Exponential Function [tex]\(f(x) = 2^x + 2\)[/tex]
Now we evaluate the exponential function at points from 0 to 8:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 3 \\
1 & 4 \\
2 & 6 \\
3 & 10 \\
4 & 18 \\
5 & 34 \\
6 & 66 \\
7 & 130 \\
8 & 258 \\
\hline
\end{array}
\][/tex]
Next, we calculate the differences between consecutive [tex]\(f(x)\)[/tex] values:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Interval} & \text{Difference} \\
\hline
1-0 & f(1) - f(0) = 4 - 3 = 1 \\
2-1 & f(2) - f(1) = 6 - 4 = 2 \\
3-2 & f(3) - f(2) = 10 - 6 = 4 \\
4-3 & f(4) - f(3) = 18 - 10 = 8 \\
5-4 & f(5) - f(4) = 34 - 18 = 16 \\
6-5 & f(6) - f(5) = 66 - 34 = 32 \\
7-6 & f(7) - f(6) = 130 - 66 = 64 \\
8-7 & f(8) - f(7) = 258 - 130 = 128 \\
\hline
\end{array}
\][/tex]
The differences for the exponential function vary, with the maximum difference being 128. Therefore, the maximum difference for the exponential function is:
[tex]\[
\max_{\text{exponential}} = 128
\][/tex]
### Conclusion
Since the maximum difference for the linear function is 2 and the maximum difference for the exponential function is 128, the exponential function [tex]\(f(x) = 2^x + 2\)[/tex] increases at the fastest rate between [tex]\(x = 0\)[/tex] and [tex]\(x = 8\)[/tex].
