https://file.aiquickdraw.com/mathbot/file/1727996122660ktil9lo3.jpg

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.

Evaluate g(f(-1)) and f(g(0)) using the graphs of f(x) and g(x).

Learn how to evaluate g(f(-1)) and f(g(0)) using function composition from graphs of f(x) and g(x). This example demonstrates key concepts in algebra, including graph interpretation and function composition. Suitable for students in Grades 9-11.

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