Let's go step by step to find each of these composite functions:

1. Find $f(g(x))$

Given:

• $f(x) = 8x + 18$
• $g(x) = x^2 + 4x - 12$

To find $f(g(x))$, substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(x^2 + 4x - 12) = 8(x^2 + 4x - 12) + 18$

Now simplify:

$f(g(x)) = 8x^2 + 32x - 96 + 18 = 8x^2 + 32x - 78$

So,

$f(g(x)) = 8x^2 + 32x - 78$

2. Find $g(f(x))$

To find $g(f(x))$, substitute $f(x) = 8x + 18$ into $g(x)$:

$g(f(x)) = g(8x + 18) = (8x + 18)^2 + 4(8x + 18) - 12$

Now expand and simplify:

$(8x + 18)^2 = 64x^2 + 288x + 324$ $4(8x + 18) = 32x + 72$

So,

$g(f(x)) = 64x^2 + 288x + 324 + 32x + 72 - 12$ $g(f(x)) = 64x^2 + 320x + 384$

3. Find $f(f(x))$

To find $f(f(x))$, substitute $f(x) = 8x + 18$ into itself:

$f(f(x)) = f(8x + 18) = 8(8x + 18) + 18$

Simplify:

$f(f(x)) = 64x + 144 + 18 = 64x + 162$

So,

$f(f(x)) = 64x + 162$

4. Find $g(g(x))$

To find $g(g(x))$, substitute $g(x) = x^2 + 4x - 12$ into itself:

$g(g(x)) = g(x^2 + 4x - 12) = (x^2 + 4x - 12)^2 + 4(x^2 + 4x - 12) - 12$

First, expand $(x^2 + 4x - 12)^2$:

$(x^2 + 4x - 12)^2 = x^4 + 8x^3 - 8x^2 - 96x + 144$

Now expand $4(x^2 + 4x - 12)$:

$4(x^2 + 4x - 12) = 4x^2 + 16x - 48$

Now simplify the expression:

$g(g(x)) = x^4 + 8x^3 - 8x^2 - 96x + 144 + 4x^2 + 16x - 48 - 12$

$g(g(x)) = x^4 + 8x^3 - 4x^2 - 80x + 84$

So,

$g(g(x)) = x^4 + 8x^3 - 4x^2 - 80x + 84$

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Do you want further details on any of these steps or explanations?

Here are 5 related questions to explore:

1. How do you find the domain of a composite function like $f(g(x))$?
2. What is the difference between $f(g(x))$ and $g(f(x))$?
3. Can a composite function ever be the same as its component functions?
4. How do you interpret the result of a composite function graphically?
5. What happens to the degree of a polynomial when composing two polynomial functions?

Tip: When composing functions, always ensure you simplify step by step to avoid errors in expansion and combining like terms.