If f(x) is an exponential function of the form of y=ab^x where f(1) = 24 and f(8) = 76, then find the value of f(11) to the nearest hundredth.

Given the exponential function \( f(x) = ab^x \) and the conditions \( f(1) = 24 \) and \( f(8) = 76 \), we need to find the value of \( f(11) \).

### Step 1: Set up the equations using the given points.
- For \( x = 1 \):
  \[
  f(1) = ab^1 = ab = 24 \quad \text{(Equation 1)}
  \]
- For \( x = 8 \):
  \[
  f(8) = ab^8 = 76 \quad \text{(Equation 2)}
  \]

### Step 2: Solve for \( b \).
Divide Equation 2 by Equation 1:
\[
\frac{ab^8}{ab} = \frac{76}{24}
\]
\[
b^7 = \frac{76}{24}
\]
\[
b^7 = \frac{76}{24} = \frac{19}{6} \approx 3.1667
\]
\[
b = \left(\frac{19}{6}\right)^{\frac{1}{7}} \approx 1.1723
\]

### Step 3: Solve for \( a \).
Substitute the value of \( b \) into Equation 1:
\[
a(1.1723) = 24
\]
\[
a = \frac{24}{1.1723} \approx 20.474
\]

### Step 4: Find \( f(11) \).
Now, use the values of \( a \) and \( b \) to calculate \( f(11) \):
\[
f(11) = ab^{11} = 20.474 \times (1.1723)^{11}
\]
Calculate \( (1.1723)^{11} \):
\[
(1.1723)^{11} \approx 2.241
\]
Thus,
\[
f(11) \approx 20.474 \times 2.241 \approx 45.91
\]

So, the value of \( f(11) \) to the nearest hundredth is approximately **45.91**.

Would you like more details on any part of this solution or have any questions?

Here are 5 related questions for further exploration:

1. How does changing the base \( b \) in an exponential function affect its graph?
2. What would the value of \( f(x) \) be if \( x \) were negative?
3. How can you determine the value of \( x \) given a specific \( f(x) \) value?
4. What is the significance of the constant \( a \) in the exponential function \( y = ab^x \)?
5. Can you derive the formula for the inverse of an exponential function?

**Tip:** When dealing with exponential functions, it's important to understand the relationship between the base and the rate of growth or decay.

Learn how to solve the exponential function f(x) = ab^x given f(1) = 24 and f(8) = 76. Detailed step-by-step solution with calculations and explanation. Suitable for high school students in Grades 10-12.

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