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Let's go through the problem one step at a time. We are given two functions:

- \( f(x) = x^2 + 4x \)
- \( g(x) = x + 6 \)

We need to find the following compositions:

### (a) \( (f \circ g)(x) \)

This means we need to substitute \( g(x) \) into \( f(x) \), so:

\[
(f \circ g)(x) = f(g(x)) = f(x + 6)
\]

Now, substitute \( x + 6 \) into \( f(x) = x^2 + 4x \):

\[
f(x + 6) = (x + 6)^2 + 4(x + 6)
\]

First, expand \( (x + 6)^2 \):

\[
(x + 6)^2 = x^2 + 12x + 36
\]

Now, distribute the 4 over \( (x + 6) \):

After expanding, we get:

\[
(f \circ g)(x) = x^2 + 16x + 60
\]

### (b) \( (g \circ f)(x) \)

This means we need to substitute \( f(x) \) into \( g(x) \), so:

\[
(g \circ f)(x) = g(f(x)) = g(x^2 + 4x)
\]

Now, substitute \( x^2 + 4x \) into \( g(x) = x + 6 \):

\[
g(x^2 + 4x) = (x^2 + 4x) + 6
\]

Simplify:

\[
g(f(x)) = x^2 + 4x + 6
\]

### (c) \( (g \circ g)(x) \)

This means we need to substitute \( g(x) \) into itself, so:

\[
(g \circ g)(x) = g(g(x)) = g(x + 6)
\]

Now, substitute \( x + 6 \) into \( g(x) = x + 6 \):

\[
g(x + 6) = (x + 6) + 6 = x + 12
\]

### Final Results:

- (a) \( (f \circ g)(x) = x^2 + 16x + 60 \)
- (b) \( (g \circ f)(x) = x^2 + 4x + 6 \)
- (c) \( (g \circ g)(x) = x + 12 \)

Do you want further details on any of these solutions?

Here are 5 related questions to explore:

1. How do you interpret compositions of functions in real-world scenarios?
2. Can you explain why function composition is not commutative?
3. What happens if we switch the roles of \( f(x) \) and \( g(x) \) for these compositions?
4. How do compositions change when one function is a constant?
5. What is the domain of a composition of functions?

**Tip**: When composing functions, always substitute one function into the other carefully, respecting the order of operations.

Given that f(x) = x^2 + 4x and g(x) = x + 6, find and simplify the compositions (f ∘ g)(x), (g ∘ f)(x), and (g ∘ g)(x).

Learn how to compute function compositions with a quadratic function f(x) = x^2 + 4x and a linear function g(x) = x + 6. This problem involves applying function composition rules and simplifying the resulting expressions. Suitable for high school students in Grades 9-12.

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