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.