We are given an exponential function of the form:

$f(x) = ab^x$

where $a$ and $b$ are constants. Additionally, we know two values of the function at specific points:

1. $f(0.5) = 12$
2. $f(7.5) = 67$

We will use this information to find the values of $a$ and $b$, and then calculate $f(14)$.

Step 1: Express the function at given points.

We know that:

1. $f(0.5) = ab^{0.5} = 12$
2. $f(7.5) = ab^{7.5} = 67$

Now we have two equations: $ab^{0.5} = 12 \quad \text{(1)}$ $ab^{7.5} = 67 \quad \text{(2)}$

Step 2: Solve for $a$ and $b$.

To eliminate $a$, divide equation (2) by equation (1): $\frac{ab^{7.5}}{ab^{0.5}} = \frac{67}{12}$ This simplifies to: $b^{7.5 - 0.5} = \frac{67}{12}$ $b^7 = \frac{67}{12}$ $b^7 = 5.5833$

Now, take the 7th root of both sides to solve for $b$: $b = \sqrt[7]{5.5833}$

Step 3: Find the value of $b$.

Let’s calculate $b$: $b \approx 1.267$

Step 4: Substitute $b$ into one of the original equations.

Now that we know $b \approx 1.267$, substitute this value back into equation (1) to solve for $a$: $a \cdot 1.267^{0.5} = 12$ $a \cdot 1.125 = 12$ $a \approx \frac{12}{1.125} = 10.67$

Step 5: Find $f(14)$.

Now we can substitute $a = 10.67$ and $b = 1.267$ into the general form of the exponential function to find $f(14)$: $f(14) = 10.67 \cdot 1.267^{14}$

Let’s calculate this value: $f(14) \approx 10.67 \cdot 1.267^{14} \approx 10.67 \cdot 6.059$ $f(14) \approx 64.6$

Final Answer:

The value of $f(14)$ to the nearest hundredth is approximately 64.60.

---

Would you like further details on any of the steps? Or perhaps more questions on exponential functions?

Here are some related questions you might find interesting:

1. How do you derive the general formula for an exponential function?
2. What would happen if the exponential function were in a different form, like $f(x) = ae^{bx}$?
3. How can you apply the concept of exponential growth to real-life scenarios, like population modeling?
4. What are the properties of exponential functions in terms of their graph and asymptotes?
5. Can you explain how to solve exponential equations using logarithms?

Tip: When dealing with exponential equations, isolating one variable at a time (like dividing equations to eliminate one variable) often simplifies solving for the constants.