Let's interpret the graph to answer your questions based on the functions $f(x)$ and $g(x)$ from the graph.

1. $(f + g)(4)$:

   • First, find $f(4)$ and $g(4)$ from the graph.
   • $f(4) = 0$ (blue point at $(4, 0)$).
   • $g(4) = 2$ (purple point at $(4, 2)$). $(f + g)(4) = f(4) + g(4) = 0 + 2 = 2.$

2. $(g - f)(3)$:

   • Find $g(3)$ and $f(3)$ from the graph.
   • $g(3) = 2$ (purple point at $(3, 2)$).
   • $f(3) = 2$ (blue point at $(3, 2)$). $(g - f)(3) = g(3) - f(3) = 2 - 2 = 0.$

3. $(g \cdot f)(1)$:

   • Find $g(1)$ and $f(1)$ from the graph.
   • $g(1) = 2$ (purple point at $(1, 2)$).
   • $f(1) = 3$ (blue point at $(1, 3)$). $(g \cdot f)(1) = g(1) \cdot f(1) = 2 \cdot 3 = 6.$

4. $\left(\frac{f}{g}\right)(-1)$:

   • Find $f(-1)$ and $g(-1)$ from the graph.
   • $f(-1) = 3$ (blue point at $(-1, 3)$).
   • $g(-1) = 2$ (purple point at $(-1, 2)$). $\left(\frac{f}{g}\right)(-1) = \frac{f(-1)}{g(-1)} = \frac{3}{2}.$

5. $g(f(0))$ (also written as $g \circ f (0)$):

   • First, find $f(0)$, then use this to find $g(f(0))$.
   • $f(0) = 3$ (blue point at $(0, 3)$).
   • Now, find $g(3) = 2$ (purple point at $(3, 2)$). $g(f(0)) = g(3) = 2.$

6. $g(f(4))$:

   • First, find $f(4)$, then use this to find $g(f(4))$.
   • $f(4) = 0$ (blue point at $(4, 0)$).
   • Now, find $g(0) = 2$ (purple point at $(0, 2)$). $g(f(4)) = g(0) = 2.$

Would you like more details or have further questions? Here are some related questions:

1. What is $f(g(2))$?
2. How do we calculate $(f - g)(0)$?
3. What would $(g \cdot f)(-2)$ be?
4. Find the value of $\left(\frac{g}{f}\right)(2)$.
5. Determine the domain of the combined function $(f + g)(x)$.

Tip: When working with composite functions like $g(f(x))$, always calculate the inner function first and then apply the result to the outer function.