https://file.aiquickdraw.com/mathbot/file/17272772098555jt79hzl.jpg

Let's work through the given problem step by step, using the two graphs provided for \( f(x) \) and \( g(x) \).

The three expressions we need to solve are:

1. \( f(g(-4)) \)
2. \( g(f(0)) \)
3. \( g(g(-8)) \)

We'll use the graphs to find the relevant values for \( f(x) \) and \( g(x) \). 

### 1. \( f(g(-4)) \)
- First, find \( g(-4) \) using the graph of \( g(x) \) (the red graph on the right).
- Then, use this value to find \( f(x) \) from the blue graph (on the left).

### 2. \( g(f(0)) \)
- First, find \( f(0) \) using the blue graph of \( f(x) \).
- Then, use this value to find \( g(x) \) from the red graph.

### 3. \( g(g(-8)) \)
- First, find \( g(-8) \) using the red graph of \( g(x) \).
- Then, find \( g \) of that result by checking the graph again.

I'll zoom in on the graphs and retrieve the exact values for these calculations now.The image is now open, and I'll analyze the graphs to find the required values.

### 1. \( f(g(-4)) \)
- Looking at the red graph of \( g(x) \), \( g(-4) \) appears to be **0**.
- Now, using the blue graph of \( f(x) \), \( f(0) \) is **-4**.

Thus, \( f(g(-4)) = f(0) = -4 \).

### 2. \( g(f(0)) \)
- From the blue graph of \( f(x) \), \( f(0) \) is **-4**.
- Now, from the red graph, \( g(-4) \) is **0**.

Thus, \( g(f(0)) = g(-4) = 0 \).

### 3. \( g(g(-8)) \)
- From the red graph, \( g(-8) \) is approximately **-2**.
- Now, using the red graph again, \( g(-2) \) is **0**.

Thus, \( g(g(-8)) = g(-2) = 0 \).

### Final answers:
1. \( f(g(-4)) = -4 \)
2. \( g(f(0)) = 0 \)
3. \( g(g(-8)) = 0 \)

Would you like any further details or explanations on how these were determined?

Here are some related questions for further exploration:
1. What is \( g(g(0)) \)?
2. Can you determine \( f(f(3)) \) from the graphs?
3. How would you calculate \( g(f(-10)) \)?
4. What does the behavior of \( g(x) \) suggest as \( x \to -\infty \)?
5. Could you approximate \( f(x) \) at \( x = 2 \)?

**Tip:** When dealing with function compositions, always start from the innermost function and work outward.

Use the figures above, which show the functions f(x) and g(x), to find the following values: 1. f(g(-4)) = ?, 2. g(f(0)) = ?, 3. g(g(-8)) = ?

Learn how to solve function compositions using graphs for f(x) and g(x). This problem involves finding the values of f(g(-4)), g(f(0)), and g(g(-8)) by interpreting the given graphs. Suitable for high school students.

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