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The image shows a graph of the exponential function \( f(x) = a^x \). The goal is to determine \( f(3) \) and \( f(-4) \) based on the graph and the given information.

From the graph, we can observe:

- \( f(0) = 1 \), because the graph passes through the point \( (0, 1) \). This is typical for an exponential function \( f(x) = a^x \), where \( f(0) = a^0 = 1 \).
- \( f(-1) = 2 \), meaning the function value at \( x = -1 \) is 2. Using the exponential formula \( f(-1) = a^{-1} = \frac{1}{a} \), we get the equation \( \frac{1}{a} = 2 \). Solving for \( a \), we find that \( a = \frac{1}{2} \).

Thus, the function is \( f(x) = \left( \frac{1}{2} \right)^x \).

Now, to find \( f(3) \) and \( f(-4) \):

1. \( f(3) = \left( \frac{1}{2} \right)^3 = \frac{1}{8} \)
2. \( f(-4) = \left( \frac{1}{2} \right)^{-4} = 2^4 = 16 \)

So, the values are:
- \( f(3) = \frac{1}{8} \)
- \( f(-4) = 16 \)

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:
1. How do we determine the base of an exponential function from a graph?
2. What happens to the function \( f(x) = a^x \) as \( a \) becomes larger or smaller?
3. How does the behavior of \( f(x) = a^x \) differ when \( a > 1 \) and \( 0 < a < 1 \)?
4. What is the significance of the point \( (0, 1) \) on exponential function graphs?
5. Can an exponential function have a negative base?

**Tip:** Exponential functions of the form \( f(x) = a^x \) grow or decay rapidly depending on whether \( a \) is greater than or less than 1.

The graph of the exponential function f(x) = a^x is given. Find f(3) and f(-4) using the graph.

Learn how to solve for f(3) and f(-4) using the graph of the exponential function f(x) = a^x. Key concepts include exponential functions, base determination, and graph interpretation. Suitable for students in Grades 10-12.

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