Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
