We are given two functions $f(x) = 4x + 4$ and $g(x) = 2x^2 + 4x$. We need to find:

1. $(f \circ g)(2)$, which means applying $g(x)$ first and then $f(x)$ to the result.
2. $(f \circ g)(x)$, the general composition of $f(x)$ and $g(x)$.

Step 1: Finding $(f \circ g)(2)$

The notation $(f \circ g)(2)$ means $f(g(2))$. So we first calculate $g(2)$, then apply $f$ to that result.

$g(x) = 2x^2 + 4x$ Substitute $x = 2$: $g(2) = 2(2)^2 + 4(2) = 2(4) + 8 = 8 + 8 = 16$

Now apply $f$ to $g(2)$: $f(x) = 4x + 4$ Substitute $x = 16$: $f(16) = 4(16) + 4 = 64 + 4 = 68$

Thus, $(f \circ g)(2) = 68$.

Step 2: Finding $(f \circ g)(x)$

The general form $(f \circ g)(x)$ means $f(g(x))$, so we substitute $g(x)$ into $f(x)$.

$f(x) = 4x + 4$ Substitute $g(x) = 2x^2 + 4x$ into $f(x)$: $f(g(x)) = 4(2x^2 + 4x) + 4$ Distribute the 4: $f(g(x)) = 8x^2 + 16x + 4$

Thus, $(f \circ g)(x) = 8x^2 + 16x + 4$.

Final Answers:

1. $(f \circ g)(2) = 68$
2. $(f \circ g)(x) = 8x^2 + 16x + 4$

Would you like further details or explanations?

Here are 5 related questions:

1. What is the result of $(g \circ f)(2)$?
2. What is $(g \circ f)(x)$?
3. How would the composition change if $f(x) = 5x + 3$?
4. What is the derivative of $(f \circ g)(x)$?
5. Can you compute $(f \circ f)(x)$?

Tip: When solving compositions of functions, always apply the innermost function first, then the outer function.