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.