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The problem involves two functions, \( f(x) = x^2 + 4x + 4 \) and \( g(x) = 2x^2 - 8 \), and it asks to determine the following composite functions:

1. \( (f \circ g)(x) \)
2. \( (g \circ f)(x) \)
3. \( (f \circ f)(x) \)
4. \( (g \circ g)(x) \)

### Let's solve each part:

#### a) \( (f \circ g)(x) = f(g(x)) \)
This means substituting \( g(x) \) into \( f(x) \):
\[
g(x) = 2x^2 - 8
\]
Now, substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = (2x^2 - 8)^2 + 4(2x^2 - 8) + 4
\]
Expand each term:
\[
(2x^2 - 8)^2 = 4x^4 - 32x^2 + 64
\]
\[
4(2x^2 - 8) = 8x^2 - 32
\]
Now, substitute back:
\[
f(g(x)) = (4x^4 - 32x^2 + 64) + (8x^2 - 32) + 4
\]
Simplify:
\[
f(g(x)) = 4x^4 - 24x^2 + 36
\]

#### b) \( (g \circ f)(x) = g(f(x)) \)
This means substituting \( f(x) \) into \( g(x) \):
\[
f(x) = x^2 + 4x + 4
\]
Now, substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = 2(x^2 + 4x + 4)^2 - 8
\]
Expand \( (x^2 + 4x + 4)^2 \):
\[
(x^2 + 4x + 4)^2 = x^4 + 8x^3 + 24x^2 + 32x + 16
\]
Now substitute:
\[
g(f(x)) = 2(x^4 + 8x^3 + 24x^2 + 32x + 16) - 8
\]
Expand and simplify:
\[
g(f(x)) = 2x^4 + 16x^3 + 48x^2 + 64x + 32 - 8
\]
\[
g(f(x)) = 2x^4 + 16x^3 + 48x^2 + 64x + 24
\]

#### c) \( (f \circ f)(x) = f(f(x)) \)
This means substituting \( f(x) \) into \( f(x) \):
\[
f(f(x)) = (x^2 + 4x + 4)^2 + 4(x^2 + 4x + 4) + 4
\]
Expand \( (x^2 + 4x + 4)^2 \) as before:
\[
(x^2 + 4x + 4)^2 = x^4 + 8x^3 + 24x^2 + 32x + 16
\]
Now, substitute back:
\[
f(f(x)) = (x^4 + 8x^3 + 24x^2 + 32x + 16) + 4(x^2 + 4x + 4) + 4
\]
Expand:
\[
4(x^2 + 4x + 4) = 4x^2 + 16x + 16
\]
Now, combine terms:
\[
f(f(x)) = x^4 + 8x^3 + 28x^2 + 48x + 36
\]

#### d) \( (g \circ g)(x) = g(g(x)) \)
This means substituting \( g(x) \) into \( g(x) \):
\[
g(g(x)) = 2(2x^2 - 8)^2 - 8
\]
Expand \( (2x^2 - 8)^2 \):
\[
(2x^2 - 8)^2 = 4x^4 - 32x^2 + 64
\]
Now, substitute back:
\[
g(g(x)) = 2(4x^4 - 32x^2 + 64) - 8
\]
Expand and simplify:
\[
g(g(x)) = 8x^4 - 64x^2 + 128 - 8
\]
\[
g(g(x)) = 8x^4 - 64x^2 + 120
\]

### Final answers:
- \( (f \circ g)(x) = 4x^4 - 24x^2 + 36 \)
- \( (g \circ f)(x) = 2x^4 + 16x^3 + 48x^2 + 64x + 24 \)
- \( (f \circ f)(x) = x^4 + 8x^3 + 28x^2 + 48x + 36 \)
- \( (g \circ g)(x) = 8x^4 - 64x^2 + 120 \)

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you compute the derivative of composite functions like \( (f \circ g)(x) \)?
2. What is the difference between \( (f \circ g)(x) \) and \( (g \circ f)(x) \)?
3. How would you solve for roots in these composite functions?
4. Can you determine the inverse function for \( f(x) \) or \( g(x) \)?
5. What happens when you take the second derivative of \( (f \circ g)(x) \)?

**Tip:** When dealing with composite functions, always substitute carefully and simplify the expression step by step to avoid errors.

Diketahui dua fungsi yaitu f(x) = x^2 + 4x + 4 dan g(x) = 2x^2 - 8. Tentukan: a) (fog)(x) b) (gof)(x) c) (fof)(x) d) (gog)(x)

Learn how to solve composite functions f(g(x)), g(f(x)), f(f(x)), and g(g(x)) with the functions f(x) = x^2 + 4x + 4 and g(x) = 2x^2 - 8. This problem involves polynomial expansion and algebraic simplification, suitable for students in Grades 10-12.

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