We are given $h(x) = \frac{64x^3 + 3x}{17 - 64x^3}$ and asked to find two functions $f(x)$ and $g(x)$ such that $(f \circ g)(x) = h(x)$. This means that:

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

We can break down the composition by identifying possible forms for $f(x)$ and $g(x)$.

Step 1: Look for a structure in $h(x)$

Notice that $h(x)$ looks like a rational function, which suggests that we could let $g(x)$ be a simpler expression inside the rational form, and $f(x)$ handle the rational part.

One approach would be to let $g(x)$ capture the polynomial expression inside the rational function, like $64x^3 + 3x$, and let $f(x)$ represent the rational operation (quotient).

Step 2: Define $g(x)$

Let: $g(x) = 64x^3$ This is a simple choice that captures the cubic term of $h(x)$.

Step 3: Define $f(x)$

Now, using $g(x) = 64x^3$, the function $f(x)$ should take this and produce the rational expression in $h(x)$. So we set: $f(x) = \frac{x + 3x}{17 - x}$

Final Answer:

Thus, we have the following functions: $g(x) = 64x^3$ $f(x) = \frac{x + 3x}{17 - x}$ Now, $(f \circ g)(x) = f(g(x)) = \frac{64x^3 + 3x}{17 - 64x^3} = h(x)$, which satisfies the requirement.

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Do you want more details on the steps, or any other questions on this?

Here are 5 related questions for deeper understanding:

1. What if we chose a different function for $g(x)$? How would that affect $f(x)$?
2. How can we confirm that the composition $(f \circ g)(x) = h(x)$ is valid for all values of $x$?
3. What is the domain of the function $h(x)$, and how does it relate to the domains of $f(x)$ and $g(x)$?
4. Can we express the same $h(x)$ with trigonometric or logarithmic functions for $f$ and $g$?
5. How would the composition change if $h(x)$ were a different type of function, like a polynomial?

Tip: When composing functions, always verify that the output of the inner function fits into the input domain of the outer function!