The graph shows an exponential function whose domain is the set of integers.

Question

Write an equation for the exponential function in the form f(x)=a(b)x.
f(x)=
Graph: (1,-3)
(2,-9)

Answer 1

To find the equation of the exponential function \( f(x) = a(b)^x \), we can use the two points provided in the graph: \( (1, -3) \) and \( (2, -9) \).

1. Substituting the first point into the equation: \[ f(1) = a(b)^1 = -3 \] This simplifies to: \[ ab = -3 \quad \text{(1)} \]

2. Substituting the second point into the equation: \[ f(2) = a(b)^2 = -9 \] This simplifies to: \[ ab^2 = -9 \quad \text{(2)} \]

Now, we have a system of equations with:

• (1) \( ab = -3 \)
• (2) \( ab^2 = -9 \)

3. Dividing equation (2) by equation (1) to solve for \( b \): \[ \frac{ab^2}{ab} = \frac{-9}{-3} \] Simplifying this, we get: \[ b = 3 \]

4. Now substitute \( b \) back into equation (1) to solve for \( a \): \[ a(3) = -3 \] This gives us: \[ a = -1 \]

5. Now we can write the exponential function: \[ f(x) = -1(3)^x \]

So, the equation for the exponential function is: \[ f(x) = -1(3)^x \]