https://file.aiquickdraw.com/mathbot/file/1725423121752pfj49dwr.jpg

Let's analyze the problem using the given graphs. The goal is to evaluate the expressions based on the two functions provided in the graphs: \( f(x) \) and \( g(x) \).

Here are the step-by-step evaluations:

1. **\( f(g(2)) \)**:
   - First, find \( g(2) \) using the graph on the right.
   - Locate \( x = 2 \) on the \( g(x) \) graph. The corresponding \( y \)-value (or \( g(2) \)) is 3.
   - Now, use this result to find \( f(3) \) on the left graph (where \( x = 3 \)).
   - The \( y \)-value at \( x = 3 \) on the \( f(x) \) graph is 1.
   - Therefore, \( f(g(2)) = f(3) = 1 \).

2. **\( g(f(4)) \)**:
   - First, find \( f(4) \) using the graph on the left.
   - Locate \( x = 4 \) on the \( f(x) \) graph. The corresponding \( y \)-value (or \( f(4) \)) is 0.
   - Now, use this result to find \( g(0) \) on the right graph (where \( x = 0 \)).
   - The \( y \)-value at \( x = 0 \) on the \( g(x) \) graph is 2.
   - Therefore, \( g(f(4)) = g(0) = 2 \).

3. **\( f(1) \)**:
   - Locate \( x = 1 \) on the \( f(x) \) graph.
   - The \( y \)-value at \( x = 1 \) is 2.
   - Therefore, \( f(1) = 2 \).

4. **\( g(5) \)**:
   - Locate \( x = 5 \) on the \( g(x) \) graph.
   - The \( y \)-value at \( x = 5 \) is -3.
   - Therefore, \( g(5) = -3 \).

### Final Answers:
- \( f(g(2)) = 1 \)
- \( g(f(4)) = 2 \)
- \( f(1) = 2 \)
- \( g(5) = -3 \)

Would you like more details on any of these steps or have further questions?

Here are 5 related questions to further your understanding:
1. How do you determine the domain and range of \( f(x) \) and \( g(x) \) from the graphs?
2. What is the composition of functions and how does it apply to real-world situations?
3. How do you find the inverse of a function using its graph?
4. Can you describe a situation where evaluating a composition of functions might be useful?
5. What changes in the graph would affect the results of \( f(g(2)) \) or \( g(f(4)) \)?

**Tip:** When evaluating compositions of functions graphically, always start with the inner function and carefully trace the corresponding value on the outer function’s graph.

Use the graphs to evaluate the expressions f(g(2)), g(f(4)), f(1), and g(5).

Learn how to evaluate expressions using function composition with graphs. This problem guides you through calculating f(g(2)), g(f(4)), f(1), and g(5) using function graphs, focusing on concepts like function composition and graph interpretation. Suitable for students in Grades 9-12.

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