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, 5, equals, 5f(5)=5 and f, of, 14, equals, 35f(14)=35, then find the value of f, of, 18f(18), to the nearest hundredth.

We are given that \( f(x) \) is an exponential function of the form \( y = ab^x \). We also know that:

\[
f(5) = 5 \quad \text{and} \quad f(14) = 35
\]

The general form of the function is \( f(x) = ab^x \).

### Step 1: Find \( a \) and \( b \)
We can use the given points to create two equations:

1. \( 5 = ab^5 \)
2. \( 35 = ab^{14} \)

### Step 2: Divide the equations to eliminate \( a \)
\[
\frac{f(14)}{f(5)} = \frac{ab^{14}}{ab^5} = b^{14-5} = b^9
\]
\[
\frac{35}{5} = b^9 \quad \Rightarrow \quad 7 = b^9
\]

Now, solve for \( b \):

\[
b = \sqrt[9]{7} \approx 1.233
\]

### Step 3: Substitute \( b \) back into one of the original equations to find \( a \)
Using \( 5 = ab^5 \):

\[
5 = a \times \left(1.233\right)^5
\]

First, calculate \( 1.233^5 \):

\[
1.233^5 \approx 2.930
\]

Then, solve for \( a \):

\[
5 = a \times 2.930 \quad \Rightarrow \quad a = \frac{5}{2.930} \approx 1.707
\]

### Step 4: Find \( f(18) \)
Now that we have \( a \) and \( b \), the function is:

\[
f(x) = 1.707 \times 1.233^x
\]

Substituting \( x = 18 \) into the equation:

\[
f(18) = 1.707 \times 1.233^{18}
\]

Calculate \( 1.233^{18} \):

\[
1.233^{18} \approx 9.896
\]

Then:

\[
f(18) \approx 1.707 \times 9.896 \approx 16.89
\]

### Final Answer:
The value of \( f(18) \) to the nearest hundredth is approximately **16.89**.

Would you like more details or have any other questions?

Here are 5 related questions:
1. How would you solve for \( b \) if the function were \( y = ab^{2x} \)?
2. Can you find \( f(x) \) for a different \( x \) value using this function?
3. What happens if \( f(14) \) was a different value?
4. How would the function change if \( b \) was negative?
5. What is the significance of the base \( b \) in exponential functions?

**Tip:** Exponential functions grow rapidly; small changes in the exponent result in large changes in the output.

Explore how to solve exponential function problems with specific values of f(5) = 5 and f(14) = 35. Understand the process of finding the function's values at different points and learn the method to calculate f(18) to the nearest hundredth. Suitable for high school students and learners interested in advanced algebraic concepts.

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