To keep everything in terms of \( x \), we need to redefine \( f \) and \( g \) such that \( f(g(x)) \) directly gives us \( H(x) = \left(2x^2 - 6\right)^{\frac{1}{3}} \). 

Given:
\[ H(x) = \left(2x^2 - 6\right)^{\frac{1}{3}} \]

We previously identified:
\[ g(x) = 2x^2 - 6 \]
\[ f(u) = u^{\frac{1}{3}} \]

However, to keep everything in terms of \( x \) while still representing \( H(x) \) as \( f(g(x)) \), we need:

\[ f(g(x)) = H(x) \]

But if we must keep all expressions in terms of \( x \), then we look at redefining \( f \) and \( g \) directly in terms of \( x \):

1. Let's consider:
\[ g(x) = 2x^2 - 6 \]
2. Then \( f(x) \) should be:
\[ f(x) = x^{\frac{1}{3}} \]

Thus, we have:
\[ f(g(x)) = f(2x^2 - 6) = (2x^2 - 6)^{\frac{1}{3}} \]

So, our functions are:
\[ f(x) = x^{\frac{1}{3}} \]
\[ g(x) = 2x^2 - 6 \]

These functions remain in terms of \( x \) and their composition yields \( H(x) \).

Would you like more details or have any questions?

Here are 8 related questions to further expand on this topic:

1. How do we handle function composition when the inner function is more complex?
2. Can you provide examples of decomposing trigonometric functions?
3. What is the relationship between function decomposition and inverse functions?
4. How do we decompose functions involving logarithms and exponentials?
5. What are the implications of function decomposition in solving differential equations?
6. How can we visualize function composition on a graph?
7. What is the significance of the order of function composition?
8. Can function decomposition help in simplifying integrals?

**Tip:** Practice decomposing functions by breaking down complex expressions into simpler parts step by step, always checking that the composition of these parts gives back the original function.

Learn how to keep functions in terms of x by understanding function composition. Explore the cube root function and its application in composing functions. Suitable for high school students studying advanced algebra and function theory.

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