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The problem in your image asks to express the given function \( h(x) = \sqrt[4]{x^2 - 9} \) as a composition of two functions, \( f \) and \( g \), such that \( h(x) = (f \circ g)(x) \), which is the same as \( h(x) = f(g(x)) \).

To solve this, you need to decompose \( h(x) \) into two functions \( f \) and \( g \), where \( g(x) \) represents the inner function and \( f(x) \) represents the outer function.

Given:
\[
h(x) = \sqrt[4]{x^2 - 9}
\]

1. **Let** \( g(x) = x^2 - 9 \) (this is the expression inside the fourth root).
   
2. **Let** \( f(x) = \sqrt[4]{x} \) (this is the outer fourth root function).

Thus, we can write:
\[
h(x) = f(g(x)) = f(x^2 - 9) = \sqrt[4]{x^2 - 9}
\]

This gives the composition:
- \( g(x) = x^2 - 9 \)
- \( f(x) = \sqrt[4]{x} \)

Would you like further details or have any questions about this solution?

Here are some related questions to deepen your understanding:

1. How would you compose functions if \( h(x) \) involved a different type of root, say a cube root?
2. What is the importance of breaking down composite functions?
3. Can you create a different composite function using different \( f(x) \) and \( g(x) \) to represent \( h(x) \)?
4. How would the composition change if the function were \( h(x) = \sqrt{x^2 - 9} \) instead?
5. What is the domain of the composed function \( h(x) \)?

**Tip:** When breaking down composite functions, always start by identifying the "inner" and "outer" parts of the expression.

Express the given function h(x) = 4√(x² - 9) as a composition of two functions f and g so that h(x) = (f ∘ g)(x).

Learn how to express the function h(x) = 4√(x² - 9) as a composition of two functions. This detailed breakdown covers function composition and radical functions, suitable for students in Grades 9-12.

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