f(x)=8x+18
  and g(x)=x2+4x−12
.

1. Find f(g(x))=
 
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2. Find g(f(x))=
 
equation editorEquation Editor

3. Find f(f(x))=
 
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4. Find g(g(x))=


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,

\[
g(g(x)) = x^4 + 8x^3 - 4x^2 - 80x + 84
\]

---

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.

f(x)=8x+18 and g(x)=x^2+4x−12. 1. Find f(g(x))= 2. Find g(f(x))= 3. Find f(f(x))= 4. Find g(g(x))=

Learn how to solve composite functions involving f(x) = 8x + 18 and g(x) = x^2 + 4x - 12 with a step-by-step solution. This covers function composition, polynomial expansion, and algebraic simplifications, suitable for students in Grades 10-12.

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