Let's go step by step through each of the problems using the given graphs for $f(x)$ and $g(x)$.

1. $f(g(-6))$

• Step 1: Start with $g(-6)$. From the graph of $g(x)$, the point $g(-6)$ corresponds to $y = 5$ (since the line passes through $(-6, 5)$).
• Step 2: Now, evaluate $f(5)$. On the graph of $f(x)$, at $x = 5$, we find $f(5) = -2$.

Thus, $f(g(-6)) = f(5) = -2$.

2. $g(f(-4))$

• Step 1: Start with $f(-4)$. From the graph of $f(x)$, the point $f(-4)$ corresponds to $y = 0$.
• Step 2: Now, evaluate $g(0)$. From the graph of $g(x)$, at $x = 0$, $g(0) = 1$.

Thus, $g(f(-4)) = g(0) = 1$.

3. $f(f(-3))$

• Step 1: Start with $f(-3)$. From the graph of $f(x)$, the point $f(-3)$ corresponds to $y = 2$.
• Step 2: Now, evaluate $f(2)$. From the graph of $f(x)$, at $x = 2$, $f(2) = -1$.

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

4. $g(g(-2))$

• Step 1: Start with $g(-2)$. From the graph of $g(x)$, the point $g(-2)$ corresponds to $y = -1$.
• Step 2: Now, evaluate $g(-1)$. From the graph of $g(x)$, at $x = -1$, $g(-1) = 4$.

Thus, $g(g(-2)) = g(-1) = 4$.

Final Answers:

1. $f(g(-6)) = -2$
2. $g(f(-4)) = 1$
3. $f(f(-3)) = -1$
4. $g(g(-2)) = 4$

Would you like further clarification or additional details on any step?

Here are 5 related questions to deepen your understanding:

1. What is $f(g(2))$?
2. How would you evaluate $g(f(1))$?
3. If $g(f(-1))$ is asked, how would you calculate it?
4. What is the result of $f(f(0))$?
5. Can you find $g(g(4))$?

Tip: When working with compositions of functions using graphs, always start by evaluating the inner function first, then use its output to evaluate the outer function.