The problem asks to express the function $H(x) = \sqrt[3]{7x^2 + 3}$ as a composition of two functions, $f(x)$ and $g(x)$, such that $(f \circ g)(x) = H(x)$. In other words, we need to find two functions where $f(g(x)) = H(x)$.

Step-by-Step Breakdown:

1. We see that $H(x) = \sqrt[3]{7x^2 + 3}$. The function involves a cube root applied to $7x^2 + 3$.
2. One option is to let the inner function $g(x)$ handle the expression inside the cube root, and let the outer function $f(x)$ apply the cube root.

So, we can define:

• $g(x) = 7x^2 + 3$ (this is the inner part inside the cube root)
• $f(x) = \sqrt[3]{x}$ (this is the cube root part)

Verification:

Now, if we compose $f(g(x))$, we get: $f(g(x)) = f(7x^2 + 3) = \sqrt[3]{7x^2 + 3}$ which is exactly $H(x)$.

Thus, the correct functions are:

$f(x) = \sqrt[3]{x}, \quad g(x) = 7x^2 + 3.$

Let me know if you need further clarification or details on this solution!

5 Related Questions:

1. What is the general method for expressing any composite function as two simpler functions?
2. How do you find the domain of a composite function?
3. Can you create different pairs of functions $f(x)$ and $g(x)$ for the same $H(x)$?
4. How do you differentiate a composite function using the chain rule?
5. What is the difference between composing functions and multiplying them?

Tip:

Always think of breaking down the function into an "inner" and "outer" function when composing functions. The outer function applies to the result of the inner function.