To solve for $(f \circ g)(x)$ and $(g \circ f)(x)$, we need to compute the composition of the functions $f(x) = 4x + 4$ and $g(x) = x - 4$.

Part 1: $(f \circ g)(x)$

By definition: $(f \circ g)(x) = f(g(x))$

1. Substitute $g(x) = x - 4$ into $f(x) = 4x + 4$: $f(g(x)) = f(x - 4)$

2. Replace $x$ in $f(x) = 4x + 4$ with $x - 4$: $f(x - 4) = 4(x - 4) + 4$

3. Simplify: $f(x - 4) = 4x - 16 + 4 = 4x - 12$

Thus: $(f \circ g)(x) = 4x - 12$

Part 2: $(g \circ f)(x)$

By definition: $(g \circ f)(x) = g(f(x))$

1. Substitute $f(x) = 4x + 4$ into $g(x) = x - 4$: $g(f(x)) = g(4x + 4)$

2. Replace $x$ in $g(x) = x - 4$ with $4x + 4$: $g(4x + 4) = (4x + 4) - 4$

3. Simplify: $g(4x + 4) = 4x + 4 - 4 = 4x$

Thus: $(g \circ f)(x) = 4x$

Final Answers:

1. $(f \circ g)(x) = 4x - 12$
2. $(g \circ f)(x) = 4x$

Would you like a detailed explanation of the steps or further clarification?

Related Questions:

1. What is the inverse of $f(x) = 4x + 4$ and $g(x) = x - 4$?
2. How does the domain and range of $(f \circ g)(x)$ differ from $f(x)$ and $g(x)$?
3. Can the functions $f(x)$ and $g(x)$ be considered bijective? Why?
4. What happens when $(f \circ g)(x)$ and $(g \circ f)(x)$ are equal?
5. How does the order of composition affect the result of $(f \circ g)(x)$ and $(g \circ f)(x)$?

Tip: Always substitute the inner function completely before simplifying in function composition to avoid errors!