Question

Question #4a7fd

Answer 1

I have maintained for several years now that it is (or should be) possible to talk about combinations of functions without mentioning an independent variable.

Explanation:

Warning for students: My position is (perhaps) non-standard
When in doubt, the pragmatic thing to do is to satisfy the person grading your work. If your grader says I'm wrong, do what the grader wants (until they are no longer grading you).

We can and do talk about the sine function and the cosine function. We can and do talk about the function formed by the quotient of two functions.

Therefore, we ought to accept statements like "the tangent (function) is the quotient of the sine and cosine (functions)."

Using the notation that leads to accepting `tan = sin/cos`.

(Many Trigonometry teachers are outraged by my saying this.)

We also talk about the function formed by the composition of functions, so we probably ought to accept "The function formed by composing the natural logarithm with itself." I prefer the notation `ln @ ln`, but that's a notational preference, not a conceptual one.

In calculus, I am quite comfortable with "the derivative of the sine function is the cosine function." Indeed this is a useful way of expressing it when teaching the chain rule.

On The Other Hand

It is difficult if not impossible to use a function to do any actual work without having an independent variable.

The sine function is of interest because it can be evaluated at some value of the domain.

Next day edit

It occurs to me that we actually do this, but usually not with named functions. The following kind of statement occurs in many treatments of the algebra of functions.

If `f` and `g` are functions, then the sum of `f` and `g`, denoted `f+g` is defined by `(f+g)(x) = f(x) + g(x)` with domain of the intersection of the domains of `f` and `g`.

For example, see Soo T. Tan Applied Calculus 9 ed p 68
and James Stewart Calculus 8 ed p 40.

Although Ron Larson's Precalculus 9 ed p 76 avoids using a phrase like "the sum, `f+g`. He does use the notation `(f+g)(x)` implying that, just as `f` is the name of the function in `f(x)`, we have `f+g` as the name of a function.