https://file.aiquickdraw.com/mathbot/file/1726457617884tpc71nz2.jpg

Let's break down and solve the problem based on the given graph.

### The problem involves:
- Two functions, \( f(x) \) (red curve) and \( g(x) \) (blue curve), whose graphs are shown.
- We are asked to find three composite function values:
  - (a) \( (g \circ f)(-1) \)
  - (b) \( (f \circ g)(4) \)
  - (c) \( (f \circ f)(4) \)

### Step 1: Find the necessary function values from the graph.

#### (a) \( (g \circ f)(-1) \)
- This means we first find \( f(-1) \), then use this result to find \( g(f(-1)) \).
  - From the graph, when \( x = -1 \), the value of \( f(x) \) is \( f(-1) = 3 \).
  - Now, find \( g(3) \) by checking the value of \( g(x) \) when \( x = 3 \) on the blue graph.
  - From the graph, \( g(3) = -3 \).
  - Therefore, \( (g \circ f)(-1) = -3 \).

#### (b) \( (f \circ g)(4) \)
- This means we first find \( g(4) \), then use this result to find \( f(g(4)) \).
  - From the graph, when \( x = 4 \), the value of \( g(x) \) is \( g(4) = 1 \).
  - Now, find \( f(1) \) by checking the value of \( f(x) \) when \( x = 1 \) on the red graph.
  - From the graph, \( f(1) = 3 \).
  - Therefore, \( (f \circ g)(4) = 3 \).

#### (c) \( (f \circ f)(4) \)
- This means we first find \( f(4) \), then use this result to find \( f(f(4)) \).
  - From the graph, when \( x = 4 \), the value of \( f(x) \) is \( f(4) = -2 \).
  - Now, find \( f(-2) \) by checking the value of \( f(x) \) when \( x = -2 \) on the red graph.
  - From the graph, \( f(-2) = 2 \).
  - Therefore, \( (f \circ f)(4) = 2 \).

### Final Answers:
- (a) \( (g \circ f)(-1) = -3 \)
- (b) \( (f \circ g)(4) = 3 \)
- (c) \( (f \circ f)(4) = 2 \)

Would you like more details or have any other questions?

Here are 5 related questions for practice:
1. What is \( (f \circ g)(0) \)?
2. Find \( g(g(2)) \).
3. What is the value of \( (g \circ f)(1) \)?
4. If \( h(x) = f(x) + g(x) \), what is \( h(2) \)?
5. Find \( f(g(-2)) \).

**Tip:** When working with composite functions, always evaluate the inner function first and then use its result to evaluate the outer function.

Let f and g be the functions whose graphs are shown below. (a) Find (g ∘ f)(−1), (b) Find (f ∘ g)(4), (c) Find (f ∘ f)(4).

Learn how to evaluate composite functions using graphs. Find (g ∘ f)(−1), (f ∘ g)(4), and (f ∘ f)(4) by interpreting the graphs of the functions f(x) and g(x). Suitable for high school students, Grades 10-12.

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