If f, of, x*f*(*x*) is an exponential function of the form of y, equals, a, b, to the power x*y*=*abx* where f, of, minus, 4, point, 5, equals, 20*f*(−4.5)=20 and f, of, 1, point, 5, equals, 11*f*(1.5)=11, then find the value of f, of, minus, 4*f*(−4), *to the nearest hundredth*.

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.

If f(x) is an exponential function of the form y = ab^x where f(-4.5) = 20 and f(1.5) = 11, then find the value of f(-4) to the nearest hundredth.

Learn how to solve an exponential function f(x) = ab^x given two points: f(-4.5) = 20 and f(1.5) = 11. This step-by-step solution covers key algebraic techniques, including setting up and solving systems of equations and applying exponentiation. Suitable for students in Grades 9-12.

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