Let's solve the given problem.

We are tasked with expressing $f(x)$ as a composition of two functions $g$ and $h$, such that $f(x) = (g \circ h)(x) = g(h(x))$.

Let’s analyze each of the functions provided to express them as compositions:

(a) $f(x) = 3\sin(x^2)$

We can break this down as follows:

• Let $h(x) = x^2$ (the inner function).
• Let $g(x) = 3\sin(x)$ (the outer function).

Thus, $f(x) = g(h(x)) = 3\sin(x^2)$.

Answer for part (a):

• $g(x) = 3\sin(x)$
• $h(x) = x^2$

(b) $f(x) = 3\sin^2(x) + 4\sin(x)$

This can be rewritten using composition:

• Let $h(x) = \sin(x)$ (the inner function).
• Let $g(x) = 3x^2 + 4x$ (the outer function, where $x$ represents $\sin(x)$).

Thus, $f(x) = g(h(x)) = 3h(x)^2 + 4h(x) = 3\sin^2(x) + 4\sin(x)$.

Answer for part (b):

• $g(x) = 3x^2 + 4x$
• $h(x) = \sin(x)$

(c) $f(x) = (1 + \sin(x^2))^3$

This can be broken down as:

• Let $h(x) = 1 + \sin(x^2)$ (inner function).
• Let $g(x) = x^3$ (outer function).

Thus, $f(x) = g(h(x)) = (1 + \sin(x^2))^3$.

Answer for part (c):

• $g(x) = x^3$
• $h(x) = 1 + \sin(x^2)$

(d) $f(x) = \sqrt{1 - \sqrt[3]{x}}$

Here’s how we can express it:

• Let $h(x) = \sqrt[3]{x}$ (inner function).
• Let $g(x) = \sqrt{1 - x}$ (outer function).

Thus, $f(x) = g(h(x)) = \sqrt{1 - \sqrt[3]{x}}$.

Answer for part (d):

• $g(x) = \sqrt{1 - x}$
• $h(x) = \sqrt[3]{x}$

Would you like further explanation on any part of this? Here are some related questions to expand your understanding:

1. How do you identify inner and outer functions when decomposing a function?
2. What are some common functions that appear frequently in function compositions?
3. How do we verify that a composition of functions equals the original function?
4. What are the applications of function compositions in real-world problems?
5. Can all functions be decomposed into compositions of simpler functions?

Tip: When identifying compositions, look for nested operations, where one function's output becomes the input of another!