Let's go through both parts (a) and (b) systematically:

(a) $g(f(-1))$

1. First, find $f(-1)$ from the graph of $f(x)$.

   • Looking at the red graph, at $x = -1$, the value of $f(x)$ is $5$.
   • So, $f(-1) = 5$.

2. Now, find $g(5)$ using the blue graph for $g(x)$.

   • At $x = 5$, the value of $g(x)$ is $0$.
   • Thus, $g(f(-1)) = g(5) = 0$.

(b) $f(g(0))$

1. First, find $g(0)$ from the graph of $g(x)$.

   • From the blue graph, at $x = 0$, the value of $g(x)$ is $-3$.
   • So, $g(0) = -3$.

2. Now, find $f(-3)$ using the red graph for $f(x)$.

   • At $x = -3$, the value of $f(x)$ is $4$.
   • Thus, $f(g(0)) = f(-3) = 4$.

Final Answers:

• $g(f(-1)) = 0$
• $f(g(0)) = 4$

Would you like more details on any of these steps?

Related Questions:

1. How do we evaluate a function at a given point using a graph?
2. What is function composition, and how does it work in problems like this?
3. Can the domain and range of $f(x)$ and $g(x)$ influence the solution to these types of problems?
4. What are some examples of real-world scenarios where function composition is used?
5. How does the behavior of $f(x)$ and $g(x)$ change as $x$ increases or decreases?

Tip:

Always ensure that you're correctly identifying the function value for the given input, especially when working with graphs, as slight misreads can lead to errors in composition problems.