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.