In composition of function f(g(x)), what do you specifically call f(x)? I know that g(x) is a domain of f(x). But how do you term f(x) in this?

Question

I'm creating math module for my students about chain rule (derivative of composition of functions) and I just got confused regarding the best terminology to better explain it using words as I do not just want to illustrate formulas in mathematic symbols, I also want to define/discuss them in written form.

I'd like to know the best terminology (or even phrase) for this:

If we have $f(x)$ and $g(x)$ and then composed these two functions by having $f(g(x))$,

1. Can I call $f(x)$ the composing function?
2. Can I call $f(x)$ the dominating function?
3. Can I call $g(x)$ the dominated function?

I want to be careful with these terms as they might be used with other topics of Calculus or other Math and could cause confusion to my students if ever they have familiarization about other Calculus/Math topics that also use these terms.

It's obvious that $g(x)$ can be called the domain of $f(x)$. But this is what I'm trying to explain in their module in written form:

If

$f(x) = \sin(x)$

$g(x) = x^3$

Then

$y = (f(g(x))$

$y = (f(g(x)) = \sin(x^3)$

Using chain rule: $y' = [f(g(x))]' \cdot g'(x) = \cos(x^3) \cdot 3x^2$ $\to$ the answer

My statement in written form of the process I did with this:

My statement in written form of the solution is:

"In our solution, by applying chain rule, we derived the {composing? ; dominating?} function $f(x)$ by using the derivative of trigonometric function and substituted $g(x)$ as the domain of the derived $f(x)$. Then, we multiplied it by the derivative of the {function f's domain? ; dominated function?} $g(x)$ by using the power rule formula since $g(x)$ is a power function."

or just simply

"In our solution, by applying chain rule, we derived the {composing? ; dominating?} function $f(x)$ by using the derivative of trigonometric function and substituted $g(x)$ as the domain of the derived $f(x)$. Then, we multiplied it by the derivative of $g(x)$ by using the power rule formula since $g(x)$ is a power function."

Feel free to recommend/suggest a better paraphrased statement if you wish. It does not have to be a complete revision of my current statement, a suggested/recommended word/phrase for what I ask should be enough.

I want my written statement to be very detailed yet direct and not confusing.

Thanks for your help!