https://file.aiquickdraw.com/mathbot/file/1726677041061au40gt22.jpg

To solve the given problem, we need to analyze the graphs of \( f(x) \) and \( g(x) \) based on the functions and perform the required calculations for compositions of functions.

### Steps to Solve:

1. **Find \( (g \circ f)(-1) \):**
   This means we first evaluate \( f(-1) \) and then plug that result into \( g(x) \).

   - From the graph, find \( f(-1) \).
   - Once \( f(-1) \) is known, evaluate \( g(f(-1)) \).

2. **Find \( (f \circ g)(1) \):**
   This means we first evaluate \( g(1) \) and then plug that result into \( f(x) \).

   - From the graph, find \( g(1) \).
   - Once \( g(1) \) is known, evaluate \( f(g(1)) \).

3. **Find \( (f \circ f)(-1) \):**
   This means we first evaluate \( f(-1) \) and then plug that result into \( f(x) \) again.

   - From the graph, find \( f(-1) \).
   - Once \( f(-1) \) is known, evaluate \( f(f(-1)) \).

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### Analysis Based on the Graph:

1. **\( f(-1) \):**
   Looking at the red curve for \( f(x) \), when \( x = -1 \), the value of \( f(-1) \) is approximately \( 2 \).
   
2. **\( g(2) \):**
   Now, find \( g(2) \) from the blue curve. When \( x = 2 \), the value of \( g(2) \) appears to be around \( -2 \).

Thus, \( (g \circ f)(-1) = g(f(-1)) = g(2) = -2 \).

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3. **\( g(1) \):**
   Looking at the blue curve for \( g(x) \), when \( x = 1 \), the value of \( g(1) \) is \( 0 \).

4. **\( f(0) \):**
   Now, find \( f(0) \) from the red curve. When \( x = 0 \), the value of \( f(0) \) is also \( 0 \).

Thus, \( (f \circ g)(1) = f(g(1)) = f(0) = 0 \).

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5. **\( f(-1) \):**
   As we saw earlier, \( f(-1) = 2 \).

6. **\( f(2) \):**
   Now, find \( f(2) \) from the red curve. When \( x = 2 \), the value of \( f(2) \) is \( 3 \).

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

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### Answers:
- **(a) \( (g \circ f)(-1) = -2 \)**
- **(b) \( (f \circ g)(1) = 0 \)**
- **(c) \( (f \circ f)(-1) = 3 \)**

Would you like further details or explanations on any step? 

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Here are 5 related questions for practice:
1. What is \( (g \circ g)(1) \)?
2. Find \( (f \circ f)(0) \).
3. What is \( g(-2) \) based on the graph?
4. Calculate \( (g \circ f)(2) \).
5. What is the value of \( f(3) \)?

**Tip:** When dealing with compositions, always start by evaluating the innermost function first!

Let f and g be the functions whose graphs are shown. Find (g∘f)(−1), (f∘g)(1), and (f∘f)(−1).

Learn how to solve function composition problems using graphical analysis for f(x) and g(x). We calculate (g∘f)(−1), (f∘g)(1), and (f∘f)(−1), using the graph of the functions. Suitable for high school students.

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