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, 0, point, 5, equals, 26f(0.5)=26 and f, of, 4, point, 5, equals, 31f(4.5)=31, then find the value of f, of, 2, point, 5f(2.5), to the nearest hundredth.

Given that \( f(x) = ab^x \) is an exponential function, and we have two points: \( f(0.5) = 26 \) and \( f(4.5) = 31 \), we need to find the value of \( f(2.5) \) to the nearest hundredth.

### Step 1: Set Up the System of Equations

We know:
\[
f(0.5) = ab^{0.5} = 26
\]
\[
f(4.5) = ab^{4.5} = 31
\]

### Step 2: Divide the Two Equations

By dividing \( f(4.5) \) by \( f(0.5) \), we eliminate \( a \) and solve for \( b \):
\[
\frac{ab^{4.5}}{ab^{0.5}} = \frac{31}{26}
\]
This simplifies to:
\[
b^{4.5 - 0.5} = \frac{31}{26}
\]
\[
b^4 = \frac{31}{26}
\]
\[
b = \left(\frac{31}{26}\right)^{\frac{1}{4}}
\]

### Step 3: Calculate the Value of \( b \)

\[
b \approx \left(\frac{31}{26}\right)^{\frac{1}{4}} \approx 1.045
\]

### Step 4: Find the Value of \( a \)

Substitute \( b \) back into the equation \( ab^{0.5} = 26 \):
\[
a \times (1.045)^{0.5} = 26
\]
\[
a \times 1.022 = 26
\]
\[
a \approx \frac{26}{1.022} \approx 25.43
\]

### Step 5: Find \( f(2.5) \)

Now, substitute \( x = 2.5 \) into the function \( f(x) = ab^x \):
\[
f(2.5) = 25.43 \times (1.045)^{2.5}
\]
\[
f(2.5) \approx 25.43 \times 1.114 \approx 28.33
\]

Thus, \( f(2.5) \approx 28.33 \) to the nearest hundredth.

Would you like further details or have any questions?

Here are five related questions:
1. How would the solution change if \( f(0.5) \) were different?
2. What is the general approach for solving exponential equations?
3. How can you verify the accuracy of the values for \( a \) and \( b \)?
4. What role does logarithmic transformation play in solving exponential equations?
5. How would you determine \( f(3) \) using the same method?

**Tip:** Always check the units and decimal places when rounding to ensure accuracy in final results.

Explore how to solve an exponential function problem with two given points, f(0.5) = 26 and f(4.5) = 31. This detailed solution involves finding the values of constants a and b and calculating f(2.5) to the nearest hundredth. Suitable for high school students studying exponential functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation