Let's define the problem step by step for better clarity. In your notation:

1. $f \circ g$ typically denotes the composition of functions $f$ and $g$, meaning $(f \circ g)(x) = f(g(x))$.
2. $f^3$ typically refers to the function $f$ applied to itself three times: $f^3(x) = f(f(f(x)))$.
3. $f \circ h$ is the composition of functions $f$ and $h$, meaning $(f \circ h)(x) = f(h(x))$.

To fully proceed with solving these or providing specific examples, I'd need the definitions of $f(x)$, $g(x)$, and $h(x)$.

Would you like to provide the specific functions for $f$, $g$, and $h$ or have any clarifications?

Here are 5 related questions for deeper understanding:

1. How do you define the functions $f$, $g$, and $h$?
2. What are the domains and ranges of the functions you're working with?
3. Do you want to expand the composition to see the results for a specific input?
4. Would you like to explore the properties of function composition (e.g., associativity)?
5. How does $f^n$ behave for higher powers or iterated function applications?

Tip: In function composition $f \circ g$, the order matters. $f(g(x))$ is generally different from $g(f(x))$.