Given the problem where $f(x)$ is an exponential function of the form $f(x) = ab^x$, with the following conditions:

• $f(-1) = 4$
• $f(5) = 25$

We need to find the value of $f(2)$ to the nearest hundredth.

Step 1: Use the function form

We know that the function has the form: $f(x) = ab^x$ where $a$ and $b$ are constants to be determined.

Step 2: Substitute given values into the function

1. For $f(-1) = 4$, substitute into the equation: $ab^{-1} = 4 \quad \Rightarrow \quad \frac{a}{b} = 4 \quad \Rightarrow \quad a = 4b$
2. For $f(5) = 25$, substitute into the equation: $ab^5 = 25$

Step 3: Solve for $a$ and $b$

Substitute $a = 4b$ into $ab^5 = 25$: $(4b)b^5 = 25 \quad \Rightarrow \quad 4b^6 = 25 \quad \Rightarrow \quad b^6 = \frac{25}{4} = 6.25$ Taking the sixth root of both sides: $b = \sqrt[6]{6.25}$ Now calculate $b$.

We find that $b \approx 1.3572$.

Step 4: Calculate $a$

Now, using $a = 4b$, we can calculate $a$: $a = 4 \times 1.3572 = 5.4288$

Step 5: Find $f(2)$

Substitute $a$ and $b$ into the original function $f(x) = ab^x$, and calculate $f(2)$: $f(2) = 5.4288 \times (1.3572)^2$

Let’s calculate $f(2)$.We find that $f(2) \approx 10.00$ to the nearest hundredth.

Thus, the value of $f(2)$ is approximately 10.00.

Would you like any further details or clarifications?

Here are 5 related questions:

1. How can we solve other types of exponential equations?
2. What are the key properties of exponential functions?
3. How does the base $b$ affect the shape of the exponential graph?
4. What role does the initial value $a$ play in an exponential function?
5. How can we apply logarithms to solve for unknowns in exponential functions?

Tip: When solving exponential functions, always pay attention to the base and initial value, as they drastically change the behavior of the function.