Let f(x)=4x+4 and g(x)=2x2+4x.
Then (f∘g)(2)=  ,
(f∘g)(x)=  .


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.

Let f(x)=4x+4 and g(x)=2x^2+4x. Then (f∘g)(2)=  , (f∘g)(x)=  .

Explore how to compose functions f(x) = 4x + 4 and g(x) = 2x^2 + 4x with detailed steps. This problem demonstrates how to calculate (f∘g)(x) and (f∘g)(2), covering key algebraic concepts in function composition. Suitable for students in Grades 9-11.

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