The image you uploaded contains graphs for the functions $f(x)$ and $g(x)$, with questions about function compositions.

Let me assist you with solving the expressions based on these graphs:

1. $f(g(-2))$: To find this, we need to first evaluate $g(-2)$ using the graph of $g(x)$, then use the value of $g(-2)$ as the input for the function $f(x)$.

2. $g(f(-3))$: We first find $f(-3)$ using the graph of $f(x)$, and then use the result as the input for $g(x)$ to find $g(f(-3))$.

3. $f(f(-1))$: We find $f(-1)$ from the graph of $f(x)$, and then substitute this value into the function $f(x)$ again to get $f(f(-1))$.

4. $g(g(0))$: First, evaluate $g(0)$ from the graph of $g(x)$, and then use that result as the input for $g(x)$ again to determine $g(g(0))$.

I’ll interpret the graphs and calculate each of these expressions for you.

Let me first analyze the graphs and provide the answers to the questions.After reviewing the graphs provided in the image:

1. For $f(g(-2))$:

   • From the graph of $g(x)$, we find $g(-2) = 4$.
   • Now, we need to evaluate $f(4)$ using the graph of $f(x)$. From the graph, $f(4) = -2$.
   • Therefore, $f(g(-2)) = -2$.

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

   • From the graph of $f(x)$, $f(-3) = -5$.
   • Now, we need to find $g(-5)$ from the graph of $g(x)$. From the graph, $g(-5) = -4$.
   • Therefore, $g(f(-3)) = -4$.

3. For $f(f(-1))$:

   • From the graph of $f(x)$, $f(-1) = 3$.
   • Now, we need to evaluate $f(3)$ using the graph of $f(x)$. From the graph, $f(3) = -1$.
   • Therefore, $f(f(-1)) = -1$.

4. For $g(g(0))$:

   • From the graph of $g(x)$, $g(0) = -1$.
   • Now, we need to evaluate $g(-1)$ using the graph of $g(x)$. From the graph, $g(-1) = -3$.
   • Therefore, $g(g(0)) = -3$.

Final Answers:

1. $f(g(-2)) = -2$
2. $g(f(-3)) = -4$
3. $f(f(-1)) = -1$
4. $g(g(0)) = -3$

Would you like a more detailed explanation or have any additional questions?

Here are some related questions to further explore the topic:

1. How do we find the composition of functions algebraically?
2. What are the differences between direct evaluation of functions and function compositions?
3. How can we use function graphs to solve more complex compositions?
4. What happens if a composition involves a point not in the domain of one of the functions?
5. How does the graph of a composition of two functions look like compared to the individual graphs?

Tip: Always double-check the graph for accuracy and ensure you’re identifying the correct function values to avoid mistakes!