https://file.aiquickdraw.com/mathbot/file/1726696173838eec5432x.jpg

Let's solve this step by step using the graph provided.

From the image, we see:

- \( f(x) \) is the **blue linear function**.
- \( g(x) \) is the **red parabolic function**.

Let's evaluate each expression:

### 1. \( (f \circ g)(2) = f(g(2)) \)
From the graph:
- Find \( g(2) \): On the graph, at \( x = 2 \), \( g(2) = -2 \).
- Now find \( f(-2) \): On the graph, at \( x = -2 \), \( f(-2) = -4 \).

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

### 2. \( (g \circ f)(2) = g(f(2)) \)
From the graph:
- Find \( f(2) \): At \( x = 2 \), \( f(2) = 2 \).
- Now find \( g(2) \): As before, \( g(2) = -2 \).

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

### 3. \( (f \circ f)(2) = f(f(2)) \)
From the graph:
- Find \( f(2) \): At \( x = 2 \), \( f(2) = 2 \).
- Now find \( f(2) \) again: At \( x = 2 \), \( f(2) = 2 \).

Thus, \( (f \circ f)(2) = 2 \).

### 4. \( (g \circ g)(2) = g(g(2)) \)
From the graph:
- Find \( g(2) \): As before, \( g(2) = -2 \).
- Now find \( g(-2) \): At \( x = -2 \), \( g(-2) = 2 \).

Thus, \( (g \circ g)(2) = 2 \).

### 5. \( (f + g)(4) = f(4) + g(4) \)
From the graph:
- Find \( f(4) \): At \( x = 4 \), \( f(4) = 4 \).
- Find \( g(4) \): At \( x = 4 \), \( g(4) = 0 \).

Thus, \( (f + g)(4) = 4 + 0 = 4 \).

### 6. \( \frac{f}{g}(2) = \frac{f(2)}{g(2)} \)
From the graph:
- Find \( f(2) \): At \( x = 2 \), \( f(2) = 2 \).
- Find \( g(2) \): At \( x = 2 \), \( g(2) = -2 \).

Thus, \( \frac{f}{g}(2) = \frac{2}{-2} = -1 \).

### Final Answers:
1. \( (f \circ g)(2) = -4 \)
2. \( (g \circ f)(2) = -2 \)
3. \( (f \circ f)(2) = 2 \)
4. \( (g \circ g)(2) = 2 \)
5. \( (f + g)(4) = 4 \)
6. \( \frac{f}{g}(2) = -1 \)

Do you want more details on any step, or have any further questions?

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Here are 5 relative questions for further exploration:
1. How do function compositions work for non-linear functions?
2. What are the algebraic forms of the given \( f(x) \) and \( g(x) \)?
3. Can you generalize the process for evaluating \( (f \circ g)(x) \) for any \( x \)?
4. How do we interpret the division of two functions graphically?
5. What happens to the composition if \( g(x) \) is not defined for certain values of \( x \)?

**Tip**: Always ensure that the values of composed functions lie within the domain of the outer function when performing function composition.

Let f be the linear function (in blue) and let g be the parabolic function (in red) below. Evaluate the following: 1. (f o g)(2), 2. (g o f)(2), 3. (f o f)(2), 4. (g o g)(2), 5. (f + g)(4), 6. (f/g)(2). Enter 'DNE' if an answer does not exist.

Explore how to perform operations like composition and addition on linear and quadratic functions, with practical examples of how to evaluate (f o g)(2), (g o f)(2), and other operations. This lesson is suitable for high school students studying advanced algebra.

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