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.