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We are given the functions \( f(x) = x + 2 \) and \( g(x) = 4x + 4 \). We are tasked with finding four composite functions and their values.

Let's go through each part step by step:

### a. \( (f \circ g)(x) \)
This represents \( f(g(x)) \), meaning we substitute \( g(x) \) into \( f(x) \).

Given:
- \( f(x) = x + 2 \)
- \( g(x) = 4x + 4 \)

Now substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(4x + 4) = (4x + 4) + 2 = 4x + 6
\]
So, \( (f \circ g)(x) = 4x + 6 \).

### b. \( (g \circ f)(x) \)
This represents \( g(f(x)) \), meaning we substitute \( f(x) \) into \( g(x) \).

Given:
- \( f(x) = x + 2 \)
- \( g(x) = 4x + 4 \)

Now substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(x + 2) = 4(x + 2) + 4 = 4x + 8 + 4 = 4x + 12
\]
So, \( (g \circ f)(x) = 4x + 12 \).

### c. \( (f \circ g)(0) \)
This represents \( f(g(0)) \).

First, find \( g(0) \):
\[
g(0) = 4(0) + 4 = 4
\]

Now substitute this into \( f(x) \):
\[
f(g(0)) = f(4) = 4 + 2 = 6
\]
So, \( (f \circ g)(0) = 6 \).

### d. \( (g \circ f)(0) \)
This represents \( g(f(0)) \).

First, find \( f(0) \):
\[
f(0) = 0 + 2 = 2
\]

Now substitute this into \( g(x) \):
\[
g(f(0)) = g(2) = 4(2) + 4 = 8 + 4 = 12
\]
So, \( (g \circ f)(0) = 12 \).

### Final Answers:
- **a.** \( (f \circ g)(x) = 4x + 6 \)
- **b.** \( (g \circ f)(x) = 4x + 12 \)
- **c.** \( (f \circ g)(0) = 6 \)
- **d.** \( (g \circ f)(0) = 12 \)

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What is \( (f \circ f)(x) \) for \( f(x) = x + 2 \)?
2. What is \( (g \circ g)(x) \) for \( g(x) = 4x + 4 \)?
3. How do you find the inverse of the function \( f(x) = x + 2 \)?
4. How does changing the slope of \( g(x) \) affect \( (f \circ g)(x) \)?
5. If \( f(x) \) were \( x^2 + 2 \), how would that change \( (f \circ g)(x) \)?

**Tip:** Always simplify inside the composite function first before applying the outer function.

For f(x) = x + 2 and g(x) = 4x + 4, find the following functions:
1. (f ∘ g)(x) 
2. (g ∘ f)(x) 
3. (f ∘ g)(0) 
4. (g ∘ f)(0)

Learn how to solve composite functions using the example f(x) = x + 2 and g(x) = 4x + 4. This problem covers function composition and algebraic concepts, providing detailed solutions for (f ∘ g)(x), (g ∘ f)(x), (f ∘ g)(0), and (g ∘ f)(0). Suitable for Grades 9-12.

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