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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].
### 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].
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