https://file.aiquickdraw.com/mathbot/file/1726629307757xg5cc7f9.jpg

Let's analyze the given graph and solve each part of the question step by step.

### Part (a): Find \( (g \circ f)(3) \)
This means we need to evaluate \( g(f(3)) \).

1. **Step 1: Find \( f(3) \)**.
   From the graph, at \( x = 3 \), the value of \( f(x) \) is \( f(3) = 2 \).

2. **Step 2: Find \( g(f(3)) = g(2) \)**.
   Now, look at the graph of \( g(x) \). At \( x = 2 \), \( g(2) = -1 \).

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

### Part (b): Find \( (f \circ g)(-3) \)
This means we need to evaluate \( f(g(-3)) \).

1. **Step 1: Find \( g(-3) \)**.
   From the graph, at \( x = -3 \), the value of \( g(x) \) is \( g(-3) = 2 \).

2. **Step 2: Find \( f(g(-3)) = f(2) \)**.
   Now, from the graph of \( f(x) \), at \( x = 2 \), \( f(2) = 3 \).

Thus, \( (f \circ g)(-3) = 3 \).

### Part (c): Find \( (f \circ f)(-1) \)
This means we need to evaluate \( f(f(-1)) \).

1. **Step 1: Find \( f(-1) \)**.
   From the graph of \( f(x) \), at \( x = -1 \), \( f(-1) = 2 \).

2. **Step 2: Find \( f(f(-1)) = f(2) \)**.
   From the graph, we know that \( f(2) = 3 \).

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

### Summary of Results:
- \( (g \circ f)(3) = -1 \)
- \( (f \circ g)(-3) = 3 \)
- \( (f \circ f)(-1) = 3 \)

Would you like a more detailed explanation for any of these steps? Feel free to ask!

Here are 5 related questions to deepen your understanding:
1. How do we evaluate composite functions in general?
2. What is the significance of the domain and range in function composition?
3. How can you tell from a graph when a function is increasing or decreasing?
4. How would you handle composite functions if the functions involved were not graphed?
5. How does the behavior of \( f \) and \( g \) change as \( x \to \infty \)?

**Tip:** When evaluating composite functions, always begin by solving the inner function first before moving on to the outer function.

Let f and g be the functions whose graphs are shown below. (a) Find (g∘f)(3), (b) Find (f∘g)(-3), (c) Find (f∘f)(-1).

Learn how to solve function composition problems using graphical representations. This detailed example covers the steps to find (g∘f)(3), (f∘g)(-3), and (f∘f)(-1). Suitable for students in Grades 10-12, this problem helps reinforce understanding of function composition and graph reading.

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