The problem in the image asks to find functions $f(x)$ and $g(x)$ such that $h(x) = (f \circ g)(x)$, where $h(x) = |9x^2 - 4|$.

This means we need to find two functions $f(x)$ and $g(x)$ such that:

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

Given $h(x) = |9x^2 - 4|$, a reasonable approach is to identify $g(x)$ as the inner function and $f(x)$ as the outer function that processes the result of $g(x)$.

Step-by-Step Solution:

1. Outer function $f(x)$: Since the outer operation is taking the absolute value, a natural choice for $f(x)$ is:

   $f(x) = |x|$

2. Inner function $g(x)$: Now, we need $g(x)$ such that when we apply $f(g(x))$, we recover $h(x) = |9x^2 - 4|$. The expression inside the absolute value is $9x^2 - 4$, so:

   $g(x) = 9x^2 - 4$

Final Answer:

$f(x) = |x| \quad \text{and} \quad g(x) = 9x^2 - 4$

When you compose these functions, you get: $(f \circ g)(x) = f(g(x)) = f(9x^2 - 4) = |9x^2 - 4|$ which matches the given $h(x)$.

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

Here are 5 related questions to deepen understanding:

1. How do you determine the composition of functions in general?
2. Why is the absolute value function appropriate for the outer function $f(x)$?
3. Can you think of alternative forms for $g(x)$ that would still satisfy $h(x)$?
4. How would the solution change if $h(x)$ didn't involve an absolute value?
5. What role does the quadratic structure of $g(x)$ play in the function composition?

Tip: When breaking down a function composition problem, start by identifying key operations (like absolute values or squares) and consider them as candidates for the outer function.