Function notation (domain "maps to" range) for a parameterized function

Question

I have a function $f(x; a)$. Note that $x \in \mathbb{R}$ and $a \in \mathbb{R}$. For simplicity, let's say the function is $f(x; a) = ax$. For my use case, it is more convenient to think about $f$ as a function of $x$ that is parameterized by $a$ (as opposed to a function of $x$ and $a$).

I'd like to express $f$ using the "maps to" function notation. My question is, is it more appropriate to express it as $f : \mathbb{R} \mapsto \mathbb{R}$ or as $f : \mathbb{R^2} \mapsto \mathbb{R}$? I can't find any examples of this notation for a function that also uses "parameterized by" notation.

My guess is the former, as $f$ is viewed as a function of one variable (that just so happens to be parameterized). However, given discussion in Why do we say function "parameterized by" vs just function of (x,y,z,...)?, there is no mathematical difference between writing $f$ as $f(x; a)$ and $f(x, a)$. However, in the latter case one would surely use the function notation $f : \mathbb{R^2} \mapsto \mathbb{R}$. This is the source of my doubt.

Answer 1

If you want to consider $a$ as a parameter, then you will write that you are using a family of functions $(f_a)_{a\in \mathbb{R}}$ : each one of them are then defined by $f_a(x) = ax$ (so $f_a:\mathbb{R} \mapsto \mathbb{R}$ for each $a\in \mathbb{R}$).

In the other case (two variables) you will indeed write $f:\mathbb{R}^2 \rightarrow \mathbb{R}, (a,x) \mapsto ax $.

But you cannot write simultaneously $f:\mathbb{R} \mapsto \mathbb{R}$ and $f:\mathbb{R}^2 \mapsto \mathbb{R}$... (the first one is not defined).

I hope it will help a bit.