If f, of, xf(x) is an exponential function of the form of y, equals, a, b, to the power xy=ab 
x
  where f, of, minus, 1, equals, 4f(−1)=4 and f, of, 5, equals, 25f(5)=25, then find the value of f, of, 2f(2), to the nearest hundredth.

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.

If f(x) is an exponential function of the form f(x) = ab^x, where f(-1) = 4 and f(5) = 25, then find the value of f(2) to the nearest hundredth.

Find the value of f(2) in the exponential function f(x) = ab^x, where f(-1) = 4 and f(5) = 25. This problem demonstrates how to solve for constants and calculate the function value to the nearest hundredth. Suitable for students in Grades 10-12.

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