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You provided two functions:

\[
f(x) = 2x + 8
\]
\[
g(x) = 4x - 8
\]

The questions ask for compositions of these functions:

### a. \((f \circ g)(x)\)
This is the composition of \(f\) and \(g\), meaning we substitute \(g(x)\) into \(f(x)\).

\[
(f \circ g)(x) = f(g(x)) = f(4x - 8)
\]
Substitute \(4x - 8\) into \(f(x)\):

\[
f(4x - 8) = 2(4x - 8) + 8
\]
Simplify:
\[
= 8x - 16 + 8 = 8x - 8
\]
So:
\[
(f \circ g)(x) = 8x - 8
\]

### b. \((g \circ f)(x)\)
This is the composition of \(g\) and \(f\), meaning we substitute \(f(x)\) into \(g(x)\).

\[
(g \circ f)(x) = g(f(x)) = g(2x + 8)
\]
Substitute \(2x + 8\) into \(g(x)\):

\[
g(2x + 8) = 4(2x + 8) - 8
\]
Simplify:
\[
= 8x + 32 - 8 = 8x + 24
\]
So:
\[
(g \circ f)(x) = 8x + 24
\]

### c. \((f \circ g \circ f)(x)\)
This is a composition of \(f\), \(g\), and \(f\), meaning we first find \(f(x)\), then \(g(f(x))\), and finally \(f(g(f(x)))\).

From part (b), we already know that:
\[
(g \circ f)(x) = 8x + 24
\]
Now substitute \(8x + 24\) into \(f(x)\):

\[
f(8x + 24) = 2(8x + 24) + 8
\]
Simplify:
\[
= 16x + 48 + 8 = 16x + 56
\]
So:
\[
(f \circ g \circ f)(x) = 16x + 56
\]

### Summary of Results:
- \((f \circ g)(x) = 8x - 8\)
- \((g \circ f)(x) = 8x + 24\)
- \((f \circ g \circ f)(x) = 16x + 56\)

Would you like more details on these compositions or have any questions?

Here are 5 follow-up questions for practice:
1. How would the results change if \(f(x) = 3x + 5\) and \(g(x) = x - 4\)?
2. What is the inverse of \(f(x) = 2x + 8\)?
3. How would you graph \((f \circ g)(x)\)?
4. What are the domains of \(f(x)\), \(g(x)\), and their compositions?
5. Can you find \((g \circ f \circ g)(x)\)?

**Tip:** When solving function compositions, always start from the innermost function and work your way outward.

Find the function compositions for f(x) = 2x + 8 and g(x) = 4x - 8: a. (f ∘ g)(x), b. (g ∘ f)(x), c. (f ∘ g ∘ f)(x)

Solve function composition problems with functions f(x) = 2x + 8 and g(x) = 4x - 8. Learn step-by-step how to find (f ∘ g)(x), (g ∘ f)(x), and (f ∘ g ∘ f)(x).

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