We are given the exponential function $f(x) = ab^x$, where $a$ and $b$ are constants. Additionally, we are provided with two specific values:

• $f(-5) = 13$
• $f(4.5) = 64$

Step 1: Use the given values to set up two equations

From $f(x) = ab^x$, we can substitute the given points to form two equations:

1. $f(-5) = 13$ becomes: $ab^{-5} = 13$

2. $f(4.5) = 64$ becomes: $ab^{4.5} = 64$

Now we have the system of equations:

1. $ab^{-5} = 13$
2. $ab^{4.5} = 64$

Step 2: Solve the system of equations

From the first equation, solve for $a$: $a = \frac{13}{b^{-5}} = 13b^5$

Substitute this expression for $a$ into the second equation: $13b^5 \cdot b^{4.5} = 64$ Simplify the equation: $13b^{9.5} = 64$ Solve for $b^{9.5}$: $b^{9.5} = \frac{64}{13} \approx 4.9231$ Now solve for $b$ by taking the 9.5th root: $b = (4.9231)^{\frac{1}{9.5}} \approx 1.366$

Step 3: Find $a$

Substitute $b = 1.366$ back into the equation $a = 13b^5$: $a = 13(1.366)^5 \approx 13 \times 5.507 \approx 71.591$

Step 4: Find $f(4)$

Now that we have values for $a$ and $b$, we can find $f(4)$: $f(4) = 71.591 \times (1.366)^4$ First, calculate $(1.366)^4$: $(1.366)^4 \approx 3.461$ Now calculate $f(4)$: $f(4) = 71.591 \times 3.461 \approx 247.82$

Final Answer:

The value of $f(4)$, to the nearest hundredth, is $\boxed{247.82}$.

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If you'd like more details or have any questions, feel free to ask!

Follow-up Questions:

1. How would you apply the same method if the function were a logarithmic one?
2. What other types of exponential functions are commonly used in modeling real-world data?
3. How can we verify our solution using a graphing calculator or software?
4. What happens if the base $b$ is less than 1 in this kind of exponential function?
5. Could we apply a similar process if we were given trigonometric function values instead?

Tip:

When solving exponential equations, always remember that the base and exponent relationship is key. Look for patterns and simplify step-by-step, and check your results by substituting back into the original function!