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We are given two functions \( f(x) \) and \( g(x) \) based on their graphs. Let’s solve each part step by step using the composition of functions.

### (a) Find \( (g \circ f)(3) \)
This means we need to first find \( f(3) \), and then substitute that value into \( g(x) \).

1. From the graph, find \( f(3) \). Looking at the graph of \( f(x) \) (red curve), when \( x = 3 \), \( f(3) = 0 \).
2. Now, find \( g(0) \). From the graph of \( g(x) \) (blue line), when \( x = 0 \), \( g(0) = -1 \).

Thus, \( (g \circ f)(3) = g(f(3)) = g(0) = -1 \).

### (b) Find \( (f \circ g)(-3) \)
This means we need to first find \( g(-3) \), and then substitute that into \( f(x) \).

1. From the graph of \( g(x) \) (blue line), when \( x = -3 \), \( g(-3) = 0 \).
2. Now, find \( f(0) \). From the graph of \( f(x) \) (red curve), when \( x = 0 \), \( f(0) = 2 \).

Thus, \( (f \circ g)(-3) = f(g(-3)) = f(0) = 2 \).

### (c) Find \( (f \circ f)(-1) \)
This means we need to first find \( f(-1) \), and then substitute that into \( f(x) \).

1. From the graph of \( f(x) \) (red curve), when \( x = -1 \), \( f(-1) = 2 \).
2. Now, find \( f(2) \). From the graph of \( f(x) \), when \( x = 2 \), \( f(2) = 3 \).

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

### Final answers:
- (a) \( (g \circ f)(3) = -1 \)
- (b) \( (f \circ g)(-3) = 2 \)
- (c) \( (f \circ f)(-1) = 3 \)

Would you like further details or clarifications?

Here are 5 related questions:
1. How do you interpret a composite function like \( (f \circ g)(x) \)?
2. What is the graphical significance of function composition?
3. How does one identify function values directly from a graph?
4. How do vertical and horizontal transformations affect the graph of a function?
5. How can you determine the inverse of a function from a graph?

**Tip:** When solving composite functions, always evaluate the innermost function first before applying the outer function.

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

Learn how to solve composite functions using graphs. This problem focuses on evaluating (g ∘ f)(3), (f ∘ g)(-3), and (f ∘ f)(-1) by interpreting the graphical representations of functions f and g. Ideal for high school students in grades 9-12.

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