https://file.aiquickdraw.com/mathbot/file/1725421547362c8wahcen.jpg

Let's evaluate the expressions using the graphs provided in the image.

### 1. \( f(g(4)) \):
   - First, find \( g(4) \) using the \( g(x) \) graph (the one on the right).
     - For \( x = 4 \), \( g(4) = 2 \).
   - Now, find \( f(2) \) using the \( f(x) \) graph (the one on the left).
     - For \( x = 2 \), \( f(2) = 4 \).
   - So, \( f(g(4)) = 4 \).

### 2. \( g(f(1)) \):
   - First, find \( f(1) \) using the \( f(x) \) graph.
     - For \( x = 1 \), \( f(1) = 2 \).
   - Now, find \( g(2) \) using the \( g(x) \) graph.
     - For \( x = 2 \), \( g(2) = 4 \).
   - So, \( g(f(1)) = 4 \).

### 3. \( f(f(0)) \):
   - First, find \( f(0) \) using the \( f(x) \) graph.
     - For \( x = 0 \), \( f(0) = 3 \).
   - Now, find \( f(3) \) using the \( f(x) \) graph.
     - For \( x = 3 \), \( f(3) = 0 \).
   - So, \( f(f(0)) = 0 \).

### 4. \( g(g(1)) \):
   - First, find \( g(1) \) using the \( g(x) \) graph.
     - For \( x = 1 \), \( g(1) = 3 \).
   - Now, find \( g(3) \) using the \( g(x) \) graph.
     - For \( x = 3 \), \( g(3) = 0 \).
   - So, \( g(g(1)) = 0 \).

### Final Answers:
1. \( f(g(4)) = 4 \)
2. \( g(f(1)) = 4 \)
3. \( f(f(0)) = 0 \)
4. \( g(g(1)) = 0 \)

Would you like more details or have any questions?

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Here are some related questions for further practice:

1. What is \( f(g(2)) \)?
2. How do you find \( g(f(3)) \) using the graphs?
3. If \( f(x) \) and \( g(x) \) were swapped, how would the answers change?
4. How can we determine if \( f(x) \) and \( g(x) \) are one-to-one functions?
5. What is the significance of the intersections of the graphs with the axes?

**Tip:** When dealing with composite functions like \( f(g(x)) \), always solve the inner function first before moving on to the outer function.

Use the graphs to evaluate the expressions: f(g(4)), g(f(1)), f(f(0)), g(g(1)).

Learn how to evaluate composite functions like f(g(4)) and g(f(1)) using graphical data. This detailed step-by-step guide helps you interpret function graphs, understand composite function evaluation, and practice key math concepts. Suitable for high school students in Grades 9-10.

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