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, 2f(−1)=2 and f, of, 2, equals, 84f(2)=84, then find the value of f, of, 2, point, 5f(2.5), to the nearest hundredth.


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

\[
f(x) = ab^x
\]

where \( a \) and \( b \) are constants, and we are also provided with the conditions:

\[
f(-1) = 2 \quad \text{and} \quad f(2) = 84.
\]

We need to find the value of \( f(2.5) \) to the nearest hundredth.

### Step 1: Use the first condition, \( f(-1) = 2 \)
Substitute \( x = -1 \) into the exponential equation:

\[
f(-1) = ab^{-1} = 2.
\]

This simplifies to:

\[
\frac{a}{b} = 2 \quad \Rightarrow \quad a = 2b.
\]

### Step 2: Use the second condition, \( f(2) = 84 \)
Substitute \( x = 2 \) into the exponential equation:

\[
f(2) = ab^2 = 84.
\]

Substitute \( a = 2b \) into this equation:

\[
(2b)b^2 = 84 \quad \Rightarrow \quad 2b^3 = 84 \quad \Rightarrow \quad b^3 = 42 \quad \Rightarrow \quad b = \sqrt[3]{42}.
\]

### Step 3: Calculate the value of \( b \)
Now, find the cube root of 42:

\[
b = \sqrt[3]{42} \approx 3.48.
\]

### Step 4: Calculate the value of \( a \)
Using \( a = 2b \) and the value of \( b \):

\[
a = 2 \times 3.48 = 6.96.
\]

### Step 5: Find \( f(2.5) \)
Now that we have \( a \) and \( b \), we can find \( f(2.5) \). Using the equation \( f(x) = ab^x \):

\[
f(2.5) = 6.96 \times (3.48)^{2.5}.
\]

Let’s calculate the value of \( (3.48)^{2.5} \).

\[
(3.48)^{2.5} \approx 22.53.
\]

Now, multiply by \( a \):

\[
f(2.5) = 6.96 \times 22.53 \approx 156.87.
\]

Thus, the value of \( f(2.5) \) to the nearest hundredth is:

\[
f(2.5) \approx 156.87.
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you derive the general form of an exponential function?
2. What is the process for solving an equation involving cube roots?
3. How would you solve for \( f(3) \) in the given equation?
4. Can you explain the relationship between logarithms and exponential functions?
5. How do you approximate powers of numbers manually?

### Tip:
When solving exponential functions, always check for additional conditions that can help in finding both the base and coefficient.

If f(x) is an exponential function of the form y = ab^x where f(-1) = 2 and f(2) = 84, then find the value of f(2.5) to the nearest hundredth.

This problem involves solving an exponential function f(x) = ab^x, where f(-1) = 2 and f(2) = 84, to find the value of f(2.5). The solution includes key concepts like exponential functions and cube roots, with a detailed step-by-step process. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation