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The problem asks to find \( (f \circ g)(x) \), which represents the composition of the functions \( f(x) \) and \( g(x) \). This means you need to substitute \( g(x) \) into \( f(x) \).

Given:
\[
f(x) = 4x + 4 \quad \text{and} \quad g(x) = 4x^2 + 4x,
\]
we compute \( (f \circ g)(x) \) as follows:

### Step 1: Substitute \( g(x) \) into \( f(x) \)
\[
f(g(x)) = f(4x^2 + 4x).
\]

### Step 2: Replace every instance of \( x \) in \( f(x) = 4x + 4 \) with \( g(x) = 4x^2 + 4x \)
\[
f(g(x)) = 4(4x^2 + 4x) + 4.
\]

### Step 3: Simplify the expression
\[
f(g(x)) = 16x^2 + 16x + 4.
\]

Thus, the simplified result is:
\[
(f \circ g)(x) = 16x^2 + 16x + 4.
\]

Would you like further clarification on any of these steps? Here are 5 related questions to expand your understanding:

1. How do you compute the composition of two functions in general?
2. Can \( (f \circ g)(x) \) always be simplified, or are there cases where it’s more complex?
3. What is the difference between \( f(g(x)) \) and \( g(f(x)) \)?
4. How does function composition differ from simple function addition or multiplication?
5. Could you compute the derivative of \( (f \circ g)(x) \) using the chain rule?

**Tip:** When performing function composition, always ensure each function's range can correctly map into the domain of the next function for accurate results.

Let f(x) = 4x + 4 and g(x) = 4x^2 + 4x. After simplifying, (f ∘ g)(x) = ?

Learn how to simplify the function composition (f ∘ g)(x) where f(x) = 4x + 4 and g(x) = 4x^2 + 4x. This problem illustrates the key steps involved in function composition, including substitution and simplification.

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