To determine whether each function represents exponential growth or exponential decay, we need to look at the base of the exponential component.
- Exponential growth occurs when the base of the exponential component is greater than 1.
- Exponential decay occurs when the base is between 0 and 1.
Let's analyze each function one by one:
1. Function [tex]\(f(x) = 485 (1.98)^t\)[/tex]:
- Here, the base is 1.98.
- Since 1.98 is greater than 1, this function represents exponential growth.
2. Function [tex]\(f(x) = \frac{1}{4} (0.18)^t\)[/tex]:
- Here, the base is 0.18.
- Since 0.18 is between 0 and 1, this function represents exponential decay.
3. Function [tex]\(f(x) = 46 (1.001)^t\)[/tex]:
- Here, the base is 1.001.
- Since 1.001 is greater than 1, this function represents exponential growth.
4. Function [tex]\(f(x) = 0.08 (0.34)^t\)[/tex]:
- Here, the base is 0.34.
- Since 0.34 is between 0 and 1, this function represents exponential decay.
Based on this analysis, we can fill in the table as follows:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline Function & Exponential growth & Exponential decay \\
\hline$f(x)=485(1.98)^t$ & \(\checkmark\) & \\
\hline$f(x)=\frac{1}{4}(0.18)^t$ & & \(\checkmark\) \\
\hline$f(x)=46(1.001)^t$ & \(\checkmark\) & \\
\hline$f(x)=0.08(0.34)^t$ & & \(\checkmark\) \\
\hline
\end{tabular}
\][/tex]
