https://file.aiquickdraw.com/mathbot/file/1726684153236uobvxsj7.jpg

Let's go step by step through each of the problems using the given graphs for \( f(x) \) and \( g(x) \).

### 1. \( f(g(-6)) \)
- **Step 1:** Start with \( g(-6) \). From the graph of \( g(x) \), the point \( g(-6) \) corresponds to \( y = 5 \) (since the line passes through \( (-6, 5) \)).
- **Step 2:** Now, evaluate \( f(5) \). On the graph of \( f(x) \), at \( x = 5 \), we find \( f(5) = -2 \).
  
Thus, \( f(g(-6)) = f(5) = -2 \).

### 2. \( g(f(-4)) \)
- **Step 1:** Start with \( f(-4) \). From the graph of \( f(x) \), the point \( f(-4) \) corresponds to \( y = 0 \).
- **Step 2:** Now, evaluate \( g(0) \). From the graph of \( g(x) \), at \( x = 0 \), \( g(0) = 1 \).

Thus, \( g(f(-4)) = g(0) = 1 \).

### 3. \( f(f(-3)) \)
- **Step 1:** Start with \( f(-3) \). From the graph of \( f(x) \), the point \( f(-3) \) corresponds to \( y = 2 \).
- **Step 2:** Now, evaluate \( f(2) \). From the graph of \( f(x) \), at \( x = 2 \), \( f(2) = -1 \).

Thus, \( f(f(-3)) = f(2) = -1 \).

### 4. \( g(g(-2)) \)
- **Step 1:** Start with \( g(-2) \). From the graph of \( g(x) \), the point \( g(-2) \) corresponds to \( y = -1 \).
- **Step 2:** Now, evaluate \( g(-1) \). From the graph of \( g(x) \), at \( x = -1 \), \( g(-1) = 4 \).

Thus, \( g(g(-2)) = g(-1) = 4 \).

### Final Answers:
1. \( f(g(-6)) = -2 \)
2. \( g(f(-4)) = 1 \)
3. \( f(f(-3)) = -1 \)
4. \( g(g(-2)) = 4 \)

Would you like further clarification or additional details on any step?

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Here are 5 related questions to deepen your understanding:
1. What is \( f(g(2)) \)?
2. How would you evaluate \( g(f(1)) \)?
3. If \( g(f(-1)) \) is asked, how would you calculate it?
4. What is the result of \( f(f(0)) \)?
5. Can you find \( g(g(4)) \)?

**Tip:** When working with compositions of functions using graphs, always start by evaluating the inner function first, then use its output to evaluate the outer function.

Use the graphs for f(x) and g(x) to evaluate the expressions.

Learn how to evaluate function compositions using graphs of f(x) and g(x). This problem includes examples such as f(g(-6)) = -2, g(f(-4)) = 1, and more. Suitable for students in Grades 9-11.

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