To find $f(x)$ and $g(x)$ such that $h(x) = (f \circ g)(x)$, we use the definition of composition:

$h(x) = f(g(x)).$

We are given: $h(x) = (3 - 9x)^2 \quad \text{and} \quad g(x) = 3 - 9x.$

Step 1: Substitute $g(x)$ into $h(x)$

$h(x) = f(g(x)) = f(3 - 9x).$

Thus: $f(3 - 9x) = (3 - 9x)^2.$

Step 2: Identify $f(u)$ where $u = g(x)$

Let $u = 3 - 9x$. Then: $f(u) = u^2.$

Final Answer:

• $g(x) = 3 - 9x$,
• $f(x) = x^2$.

These satisfy the given composition $h(x) = (f \circ g)(x)$.

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Would you like a deeper explanation or a graph of these functions?
Here are 5 related questions to explore:

1. What is the domain of $f(x)$ and $g(x)$?
2. How does the composition of functions work conceptually?
3. What are examples of other $h(x)$ functions and their decompositions?
4. How does this apply to real-world problems, such as nested relationships?
5. Can we reverse the process to find $g(x)$ given $f(x)$ and $h(x)$?

Tip: Always check the domain and range of each function in composition problems to ensure they are compatible.