TITLE: Why do we say function "parameterized by" vs just function of (x,y,z,...)?
QUESTION [28 upvotes]: I'm studying statistics, and in a lot of textbooks, the regression formulas always refers to the functions themselves as f(x) parameterized by a,b,c or something. And they are often written $f(x; a,b,c)$, why can't we just write $f(x, a, b, c)$?
I would like to know perhaps the history of how this came about, and why there is this seemingly dual method of expressing the same idea. Is there some esoteric mathematical notation that a noob like me hasn't encountered yet, or did mathematicians of old just like doing things in different ways?
An example: A function that maps age of tree to height:
$$f(age; growth\_rate) = age * growth\_rate$$ vs $$f(age, growth\_rate) = age * growth\_rate$$

REPLY [7 votes]: Think to something like this: $f(x;\sigma) = e^{-x^2/\sigma^2}$.
If you read for example "the derivative of $f$" probably you immediately undestand that we are speaking of $\partial f/\partial x$ because you recognize $f$ as a function of $x$, whose definition contains some $\sigma$ which is not a "variable".
Otherwise if you read:
$f(x, \sigma) = e^{-x^2/\sigma^2}$
a sentence like "the derivative of $f$" would be ambiguous: it's $\partial f/\partial x$? $\partial f/\partial \sigma$? $\nabla f = (\partial f/\partial x, \partial f/\partial \sigma)$?
So without this distinction every time one has to write explicitly "the derivative of $f$ with respect to the variable $x$".