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Given f(x), find g(x) and h(x) such that f(x)=g(h(x))
 and neither g(x) nor h(x) is solely x.

f(x)=−2x3+4⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√+2

To find functions \( g(x) \) and \( h(x) \) such that \( f(x) = g(h(x)) \) and neither \( g(x) \) nor \( h(x) \) is just \( x \), we need to analyze the given function:

\[
f(x) = \sqrt{-2x^3 + 4} + 2
\]

### Step 1: Identify \( h(x) \)
We want to identify an inner function \( h(x) \) such that the expression inside the square root is represented by \( h(x) \). Let’s set:
\[
h(x) = -2x^3 + 4
\]

### Step 2: Identify \( g(x) \)
With \( h(x) = -2x^3 + 4 \), we can rewrite \( f(x) \) in terms of \( g(h(x)) \). Since \( f(x) = \sqrt{h(x)} + 2 \), let’s define:
\[
g(x) = \sqrt{x} + 2
\]

### Verification
Now, substituting \( h(x) = -2x^3 + 4 \) into \( g(h(x)) \):
\[
g(h(x)) = g(-2x^3 + 4) = \sqrt{-2x^3 + 4} + 2 = f(x)
\]
This matches \( f(x) \), so the functions are correct.

### Final Answer
\[
g(x) = \sqrt{x} + 2
\]
\[
h(x) = -2x^3 + 4
\]

Would you like further details or have any questions?

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Here are 5 related questions to consider:

1. How would \( f(x) \) change if the constant outside the square root in \( g(x) \) was different?
2. What if we used a different exponent for \( x \) in \( h(x) \); how would that affect \( g(x) \)?
3. Can we choose multiple different \( g(x) \) and \( h(x) \) pairs for the same \( f(x) \)?
4. What types of functions work well with function composition, and why?
5. How does function composition affect the domain and range of the resulting function?

**Tip:** When breaking down complex functions, start by identifying "outer" and "inner" parts to simplify composition.

Given f(x), find g(x) and h(x) such that f(x) = g(h(x)) and neither g(x) nor h(x) is solely x.

f(x) = √(-2x^3 + 4) + 2

Learn how to express f(x) = √(-2x^3 + 4) + 2 as a composition of functions g(x) and h(x), where neither function is solely x. This problem explores function composition and polynomial transformations, ideal for students in Grades 10-12.

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