Decompose the function f, of, g, of, h, of, x, equals, 8, sine, x, plus, 2f(g(h(x)))=8sinx+2 into f, of, xf(x), g, of, xg(x) and h, of, xh(x).

To decompose the function \( f(g(h(x))) = 8\sin(x) + 2 \) into individual functions \( f(x) \), \( g(x) \), and \( h(x) \), let's go through it step-by-step.

We have:
\[
f(g(h(x))) = 8\sin(x) + 2
\]

Our goal is to find functions \( f \), \( g \), and \( h \) such that when composed as \( f(g(h(x))) \), we obtain the expression \( 8\sin(x) + 2 \).

### Step 1: Identify the innermost function \( h(x) \)
Let's set:
\[
h(x) = x
\]
This simplifies our problem because \( h(x) \) will simply return \( x \) for any \( x \).

### Step 2: Define \( g(x) \) such that \( g(h(x)) = g(x) \) transforms the expression.
To approach the form \( 8\sin(x) + 2 \), let's set:
\[
g(x) = 8\sin(x)
\]
This means that \( g(h(x)) = 8\sin(x) \), which closely resembles the expression \( 8\sin(x) + 2 \) except for the \( +2 \) term.

### Step 3: Define \( f(x) \) to adjust the expression as needed
Now we need \( f(x) \) such that:
\[
f(g(x)) = f(8\sin(x)) = 8\sin(x) + 2
\]
To achieve this, we can set:
\[
f(x) = x + 2
\]

### Summary
The functions \( f(x) \), \( g(x) \), and \( h(x) \) are as follows:
- \( h(x) = x \)
- \( g(x) = 8\sin(x) \)
- \( f(x) = x + 2 \)

Now, if we compose these functions:
\[
f(g(h(x))) = f(g(x)) = f(8\sin(x)) = 8\sin(x) + 2
\]
This matches the original expression, so our decomposition is correct.

Would you like further explanation on any step, or have questions?

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Here are 5 related questions to deepen understanding:
1. How would the decomposition change if the function had a different additive constant?
2. Can we find alternative decompositions with the same final function?
3. How does the decomposition of trigonometric functions affect the composition?
4. What is the impact of choosing a different innermost function \( h(x) \) in decomposition?
5. How does this approach generalize to other functions besides trigonometric ones?

**Tip:** In function decomposition, start by simplifying the innermost function first to build up to the target expression step-by-step.

Decompose the function f(g(h(x))) = 8sin(x) + 2 into f(x), g(x), and h(x).

Explore the step-by-step decomposition of the composite function f(g(h(x))) = 8sin(x) + 2 into individual functions f(x), g(x), and h(x). Learn key concepts in function composition and trigonometry, suitable for high school students in Grades 10-12.

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