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To find the exponential equation of the form \( f(x) = a \cdot e^{bx} \) from the given data, we will analyze the values of \(x\) and \(f(x)\) provided in the table.

### Data provided:
\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 3 & 6 & 9 & 15 & 21 \\
\hline
f(x) & 100 & 103.3 & 106.709 & 113.868 & 121.507 \\
\hline
\end{array}
\]

### Steps to find the equation:
1. **Identify the general form**: The equation is assumed to have the exponential form:
   \[
   f(x) = a \cdot e^{bx}
   \]
   where \(a\) and \(b\) are constants we need to find.

2. **Linearize the equation**: Taking the natural logarithm (ln) of both sides of the equation:
   \[
   \ln(f(x)) = \ln(a \cdot e^{bx}) = \ln(a) + bx
   \]
   This equation is now linear in terms of \( \ln(f(x)) \), which means we can treat this as a linear regression problem to determine \(a\) and \(b\).

3. **Perform a regression**: We will use the given data points for \(x\) and \(f(x)\) to compute a best fit using logarithmic transformation.

I'll now perform the necessary calculations to find \(a\) and \(b\).The exponential equation that best fits the given data is:

\[
f(x) = 96.806 \cdot e^{0.01082x}
\]

Here:
- \(a = 96.806\)
- \(b = 0.01082\)

This represents the exponential relationship between \(x\) and \(f(x)\).

Would you like further details or have any questions about this solution?

### Related Questions:
1. How can we verify if this equation accurately fits the data?
2. What does the constant \(b\) represent in an exponential function?
3. Can this model be used for predicting values of \(f(x)\) beyond the given range of \(x\)?
4. How would this model change if the growth rate were negative?
5. How can you apply this equation to practical problems, such as population growth?

**Tip:** When dealing with exponential functions, always check the value of \(b\). If \(b > 0\), the function represents growth; if \(b < 0\), it represents decay.

Find the exponential equation for the given data points.

This problem demonstrates how to find the exponential equation f(x) = a * e^(bx) using regression analysis on the provided data points. Given values of x and f(x) are analyzed, and the equation f(x) = 96.806 * e^(0.01082x) is derived. Suitable for high school students studying exponential functions and regression analysis.

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