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.