If f, of, xf(x) is an exponential function of the form of y, equals, a, b, to the power xy=ab 
x
  where f, of, 3, point, 5, equals, 8f(3.5)=8 and f, of, 7, equals, 33f(7)=33, then find the value of f, of, 6, point, 5f(6.5), to the nearest hundredth.

Given the problem, we are tasked with finding the exponential function \( f(x) = ab^x \) based on the provided points \( f(3.5) = 8 \) and \( f(7) = 33 \). We then need to determine \( f(6.5) \) to the nearest hundredth.

### Step 1: Set up the system of equations
Using the general exponential form \( f(x) = ab^x \), we can set up two equations based on the information given:

1. \( f(3.5) = 8 \), so:
   \[
   ab^{3.5} = 8
   \]

2. \( f(7) = 33 \), so:
   \[
   ab^7 = 33
   \]

### Step 2: Solve for \( a \) and \( b \)
To eliminate \( a \), divide the second equation by the first:

\[
\frac{ab^7}{ab^{3.5}} = \frac{33}{8}
\]
This simplifies to:
\[
b^{7 - 3.5} = \frac{33}{8}
\]
\[
b^{3.5} = \frac{33}{8} \approx 4.125
\]

Next, solve for \( b \) by taking the logarithm of both sides:
\[
b = \left( 4.125 \right)^{1/3.5}
\]
Using a calculator:
\[
b \approx 1.426
\]

### Step 3: Solve for \( a \)
Substitute the value of \( b \) back into one of the original equations, such as \( ab^{3.5} = 8 \):
\[
a(1.426)^{3.5} = 8
\]
Calculate \( (1.426)^{3.5} \):
\[
1.426^{3.5} \approx 2.851
\]
Now solve for \( a \):
\[
a \times 2.851 = 8
\]
\[
a \approx \frac{8}{2.851} \approx 2.806
\]

### Step 4: Find \( f(6.5) \)
Now that we have the values of \( a \) and \( b \), we can find \( f(6.5) \):
\[
f(6.5) = 2.806 \times (1.426)^{6.5}
\]
First, calculate \( 1.426^{6.5} \):
\[
1.426^{6.5} \approx 6.162
\]
Now, multiply by \( a \):
\[
f(6.5) \approx 2.806 \times 6.162 \approx 17.29
\]

### Final Answer:
The value of \( f(6.5) \) is approximately **17.29**.

---

Do you want further details or have any questions?

Here are 5 relative questions for further exploration:
1. How do exponential functions differ from linear functions?
2. What are the properties of the base \( b \) in an exponential function?
3. How does logarithmic scaling relate to solving exponential equations?
4. Could the exponential model be used in real-world growth scenarios? 
5. What is the significance of base \( e \) in natural exponential functions?

**Tip:** Always check your calculations by plugging values back into the original equations to verify consistency!

If f(x) is an exponential function of the form y = ab^x where f(3.5) = 8 and f(7) = 33, then find the value of f(6.5) to the nearest hundredth.

Find the value of f(6.5) for an exponential function f(x) = ab^x where f(3.5) = 8 and f(7) = 33. This solution explains key exponential function concepts, the use of logarithms, and solving a system of equations. Suitable for high school students.

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