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

$f(x) = ab^x$

with the following conditions:

• $f(-4.5) = 20$
• $f(1.5) = 11$

We are asked to find the value of $f(-4)$, rounded to the nearest hundredth.

Step 1: Use the exponential form

We know that:

$f(x) = ab^x$

Substitute the given values for $f(x)$:

$f(-4.5) = ab^{-4.5} = 20$ $f(1.5) = ab^{1.5} = 11$

Step 2: Set up a system of equations

From the two conditions, we have:

$ab^{-4.5} = 20 \quad \text{(Equation 1)}$ $ab^{1.5} = 11 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

To eliminate $a$, divide Equation 2 by Equation 1:

$\frac{ab^{1.5}}{ab^{-4.5}} = \frac{11}{20}$

Simplifying the left-hand side:

$b^{1.5 - (-4.5)} = \frac{11}{20}$ $b^6 = \frac{11}{20}$

Now, solve for $b$ by taking the 6th root of both sides:

$b = \left( \frac{11}{20} \right)^{\frac{1}{6}}$

Step 4: Calculate $b$

Let’s compute the value of $b$.The value of $b$ is approximately $0.905$.

Step 5: Find $a$

Now that we know $b$, we can substitute it back into one of the original equations to solve for $a$. Using Equation 1:

$ab^{-4.5} = 20$

Substitute $b \approx 0.905$:

$a(0.905)^{-4.5} = 20$

Solve for $a$:

$a = \frac{20}{(0.905)^{-4.5}}$

Let’s compute the value of $a$.The value of $a$ is approximately $12.77$.

Step 6: Calculate $f(-4)$

Now that we have both $a$ and $b$, we can calculate $f(-4)$. Using the formula:

$f(x) = ab^x$

Substitute $a \approx 12.77$, $b \approx 0.905$, and $x = -4$:

$f(-4) = 12.77 \times (0.905)^{-4}$

Let’s compute $f(-4)$.The value of $f(-4)$ is approximately $19.03$, rounded to the nearest hundredth.

Would you like further details or have any questions?

Here are five related questions you might explore:

1. How would the values of $a$ and $b$ change if different points were given?
2. Can you find $f(0)$ using the same values of $a$ and $b$?
3. What is the general behavior of exponential functions as $x$ increases or decreases?
4. How does the base $b$ affect the growth or decay of the function?
5. Could you solve the system of equations if $f(x)$ were a logarithmic function instead?

Tip: Exponential functions with $b < 1$ represent decay, while $b > 1$ represent growth.