https://file.aiquickdraw.com/mathbot/file/1726456696987dh0vbxxv.jpg

Let's work through the problem together by analyzing the graph and solving each part:

We have two functions, \( f(x) \) and \( g(x) \), represented by the red and blue curves, respectively. The question involves evaluating composite functions.

### (a) Find \( (g \circ f)(-1) \)
This is the composition of the functions, meaning we first evaluate \( f(-1) \), then use that result to find \( g \) at that value.

1. Find \( f(-1) \): 
   From the graph, when \( x = -1 \), the value of \( f(x) \) (red curve) is \( 1 \). So, \( f(-1) = 1 \).
   
2. Now, find \( g(1) \): 
   From the graph, when \( x = 1 \), the value of \( g(x) \) (blue curve) is \( -2 \). Therefore, \( g(1) = -2 \).

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

### (b) Find \( (f \circ g)(4) \)
This is also a composition, meaning we first evaluate \( g(4) \), then use that result to find \( f \) at that value.

1. Find \( g(4) \): 
   From the graph, when \( x = 4 \), the value of \( g(x) \) (blue curve) is \( -3 \). So, \( g(4) = -3 \).
   
2. Now, find \( f(-3) \): 
   From the graph, when \( x = -3 \), the value of \( f(x) \) (red curve) is \( 2 \). Therefore, \( f(-3) = 2 \).

Thus, \( (f \circ g)(4) = f(g(4)) = f(-3) = 2 \).

### (c) Find \( (f \circ f)(4) \)
This composition involves evaluating \( f(4) \), then using that result to find \( f \) at that value.

1. Find \( f(4) \): 
   From the graph, when \( x = 4 \), the value of \( f(x) \) (red curve) is \( 5 \). So, \( f(4) = 5 \).
   
2. Now, find \( f(5) \): 
   From the graph, when \( x = 5 \), the value of \( f(x) \) (red curve) is also \( 5 \). Therefore, \( f(5) = 5 \).

Thus, \( (f \circ f)(4) = f(f(4)) = f(5) = 5 \).

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### Summary of answers:
- **(a)** \( (g \circ f)(-1) = -2 \)
- **(b)** \( (f \circ g)(4) = 2 \)
- **(c)** \( (f \circ f)(4) = 5 \)

Would you like further explanation on any of the steps or additional details?

### Related Questions:
1. What is the value of \( (f \circ g)(-1) \)?
2. Find \( (g \circ g)(2) \).
3. How does the composition of functions differ from basic function evaluation?
4. Can you explain why the graph of a function helps in determining the composition?
5. How would you solve \( (f \circ g)(-4) \)?

### Tip:
When evaluating compositions, always work inside out—start with the innermost function first and use its result for 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).

Solve function composition problems by analyzing the graphs of f(x) and g(x). Learn how to find (g ∘ f)(-1), (f ∘ g)(4), and (f ∘ f)(4) using function composition rules and graph interpretation.

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