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?

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.