Question

which exponential function has a growth factor of 5? \

$$\begin{tabular}{|c|c|} \\hline \\( x \\) & \\( f(x) \\) \\\\ \\hline -2 & \\( \\frac{1}{8} \\) \\\\ \\hline -1 & \\( \\frac{1}{4} \\) \\\\ \\hline 0 & \\( \\frac{1}{2} \\) \\\\ \\hline 1 & 1 \\\\ \\hline 2 & 2 \\\\ \\hline \\end{tabular}$$

\\( f(x) = 0.5(2^x) \\) graph of an exponential function

Explanation:

Step1: Recall the exponential function form

The general form of an exponential function is \( f(x) = a(b^x) \), where \( b \) is the growth factor (for \( b>1 \)) or decay factor (for \( 0 < b < 1 \)). We need to find which function has \( b = 5 \).

Step2: Analyze the table

For the table, let's find the ratio between consecutive \( f(x) \) values. From \( x=-2 \) to \( x=-1 \): \( \frac{\frac{1}{4}}{\frac{1}{8}} = 2 \). From \( x=-1 \) to \( x=0 \): \( \frac{\frac{1}{2}}{\frac{1}{4}} = 2 \). From \( x=0 \) to \( x=1 \): \( \frac{1}{\frac{1}{2}} = 2 \). So the growth factor here is 2.

Step3: Analyze the function \( f(x) = 0.5(2^x) \)

In this function, the base of the exponent is 2, so the growth factor is 2.

Step4: Analyze the graph (assuming it's a decay graph, but wait, maybe we missed a function? Wait, the problem might have a typo or missing options? Wait, no, maybe the original problem had more options. Wait, the user's image shows three parts, but maybe the fourth part (not shown) has a function with growth factor 5. Wait, maybe the user missed pasting all options. But assuming the standard, an exponential function with growth factor 5 would be of the form \( f(x) = a(5^x) \). For example, if there was a function like \( f(x) = 3(5^x) \), its growth factor is 5.

Wait, maybe the original problem's options were cut. But based on the given, the table has growth factor 2, the function \( f(x)=0.5(2^x) \) has growth factor 2, the graph is a decay (so base <1). So likely, the correct function (not shown here) would be one with \( b = 5 \) in \( f(x)=a(b^x) \).

But since the user's problem might have a missing option, but following the process, the growth factor is the base of the exponential term. So a function like \( f(x) = 2(5^x) \) would have growth factor 5.

Answer:

A function of the form \( f(x) = a(5^x) \) (e.g., \( f(x) = 3(5^x) \)) has a growth factor of 5. (Note: Since the provided options' growth factors are 2, the correct option with growth factor 5 was likely missed in the pasted content.)