We are given that the function $f(x)$ is an exponential function of the form:

$f(x) = ab^x$

Where $a$ and $b$ are constants, and we are provided with the following information:

• $f(-5) = 11$
• $f(0) = 30$

Step 1: Use $f(0) = 30$ to find $a$

At $x = 0$, the exponential function becomes: $f(0) = ab^0 = a$ Since $f(0) = 30$, we know: $a = 30$

Step 2: Use $f(-5) = 11$ to find $b$

Now substitute $f(-5) = 11$ and $a = 30$ into the equation: $f(-5) = ab^{-5} = 30b^{-5} = 11$ Solve for $b$: $b^{-5} = \frac{11}{30}$ $b = \left( \frac{11}{30} \right)^{-\frac{1}{5}} = \left( \frac{30}{11} \right)^{\frac{1}{5}}$

Calculating $b$: $b \approx 1.219$

Step 3: Find $f(-3.5)$

Now that we have $a = 30$ and $b \approx 1.219$, we can find $f(-3.5)$: $f(-3.5) = 30 \cdot (1.219)^{-3.5}$

Calculating this: $f(-3.5) \approx 30 \cdot (0.8203) \approx 24.609$

Thus, the value of $f(-3.5)$ to the nearest hundredth is: $\boxed{24.61}$

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Would you like more details or have any questions?

Here are five questions for further exploration:

1. How do you derive the general form of an exponential function?
2. What happens to $f(x)$ as $x$ becomes very large in exponential functions?
3. How would you graph $f(x) = ab^x$?
4. Can exponential functions have negative bases? Why or why not?
5. What is the significance of the base $b$ in an exponential function?

Tip: In exponential functions, when $b > 1$, the function grows, and when $0 < b < 1$, the function decays.