Answer :
To determine which exponential function has a growth factor of 5, we need to understand the concept of exponential growth. In an exponential function of the form [tex]\(f(x) = a(b^x)\)[/tex], the base [tex]\(b\)[/tex] is the growth factor. It determines how the function grows as [tex]\(x\)[/tex] increases.
Let's examine each given function:
1. [tex]\(f(x) = 2(5^x)\)[/tex]
Here, the base of the exponent is 5. Therefore, the growth factor is 5.
2. [tex]\(f(x) = 0.5(2^x)\)[/tex]
In this function, the base of the exponent is 2. Thus, the growth factor is 2.
Next, we have a table of values to consider:
[tex]\[ \begin{tabular}{|c|c|} \hline x & f(x) \\ \hline -2 & \frac{1}{8} \\ \hline -1 & 1 \\ \hline \end{tabular} \][/tex]
However, the values in the table are specific outputs at given [tex]\(x\)[/tex] values and do not directly relate to identifying the growth factor of the given functions.
From our analysis:
- The function [tex]\(f(x) = 2(5^x)\)[/tex] has a growth factor of 5.
- The function [tex]\(f(x) = 0.5(2^x)\)[/tex] has a growth factor of 2.
Therefore, the correct function with a growth factor of 5 is:
[tex]\[ f(x) = 2(5^x) \][/tex]
This corresponds to the first function.
Let's examine each given function:
1. [tex]\(f(x) = 2(5^x)\)[/tex]
Here, the base of the exponent is 5. Therefore, the growth factor is 5.
2. [tex]\(f(x) = 0.5(2^x)\)[/tex]
In this function, the base of the exponent is 2. Thus, the growth factor is 2.
Next, we have a table of values to consider:
[tex]\[ \begin{tabular}{|c|c|} \hline x & f(x) \\ \hline -2 & \frac{1}{8} \\ \hline -1 & 1 \\ \hline \end{tabular} \][/tex]
However, the values in the table are specific outputs at given [tex]\(x\)[/tex] values and do not directly relate to identifying the growth factor of the given functions.
From our analysis:
- The function [tex]\(f(x) = 2(5^x)\)[/tex] has a growth factor of 5.
- The function [tex]\(f(x) = 0.5(2^x)\)[/tex] has a growth factor of 2.
Therefore, the correct function with a growth factor of 5 is:
[tex]\[ f(x) = 2(5^x) \][/tex]
This corresponds to the first function.
