To solve the given problems using the provided graph of the functions $f(x)$ and $g(x)$, let's break down each expression:

1. $(f + g)(3)$

This is asking for the sum of the values of $f(x)$ and $g(x)$ when $x = 3$.

• From the graph, $f(3)$ is the red point at $x = 3$, which corresponds to the value 2.
• Similarly, $g(3)$ is the blue point at $x = 3$, which corresponds to the value -3.

Thus, $(f + g)(3) = f(3) + g(3) = 2 + (-3) = -1$.

2. $(g - f)(-2)$

This is asking for the difference between the values of $g(x)$ and $f(x)$ when $x = -2$.

• From the graph, $g(-2)$ is the blue point at $x = -2$, which corresponds to the value 2.
• $f(-2)$ is the red point at $x = -2$, which corresponds to the value 3.

Thus, $(g - f)(-2) = g(-2) - f(-2) = 2 - 3 = -1$.

3. $(f \cdot g)(5)$

This is asking for the product of the values of $f(x)$ and $g(x)$ when $x = 5$.

• From the graph, $f(5)$ is the red point at $x = 5$, which corresponds to the value -2.
• $g(5)$ is the blue point at $x = 5$, which corresponds to the value -4.

Thus, $(f \cdot g)(5) = f(5) \cdot g(5) = (-2) \cdot (-4) = 8$.

4. $(g \cdot f)(-4)$

This is asking for the product of the values of $g(x)$ and $f(x)$ when $x = -4$.

• From the graph, $g(-4)$ is the blue point at $x = -4$, which corresponds to the value -3.
• $f(-4)$ is the red point at $x = -4$, which corresponds to the value 4.

Thus, $(g \cdot f)(-4) = g(-4) \cdot f(-4) = (-3) \cdot 4 = -12$.

Summary of results:

1. $(f + g)(3) = -1$
2. $(g - f)(-2) = -1$
3. $(f \cdot g)(5) = 8$
4. $(g \cdot f)(-4) = -12$

Would you like more details or have any other questions?

Here are 5 relative questions to deepen understanding:

1. What is the value of $(f + g)(-1)$?
2. How do you interpret the intersection point of $f(x)$ and $g(x)$?
3. Can you find $(g \cdot f)(1)$?
4. What happens when $(f - g)(0)$?
5. How would you calculate $f(4) + g(4)$?

Tip: Always check both the function's x-coordinate and y-coordinate values carefully before solving function-related problems from graphs.