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.