Notation and terminology for functions, interpreting $f(y)$

Question

It seems to me there are two different interpretations of a symbol $f(y)$. I will explain what I mean:

Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example). Also suppose I have a dependence between $x$ and $y$ which is another variable. Say $x = 2y$.

I seems to me I can interpret $f$ and particularly the symbol $f(y)$ in two different ways:

1. $f$ is strictly a map and it taken whatever variable we give it and maps it accordingly. In this case $f(y) = y$ as $f$ is the identity map so just maps $y$ to itself.

2. $f$ is a variable dependent on $x$, since I've defined $f(x) = x$, then when I denote $f(y)$ I can interpret it as the variable $f$ but now expressed in terms of $y$ instead of in terms of $x$, so in this case $f(y) = x = 2y$. In this case I viewed $f$ as already being defined in terms of x and the symbol $f(y)$ merely gives me this predefined variable $f$ in terms of $y$.

My question is what is the usual interpretation, and if there is convenient notation and/or common terminology which distinguish the two.

Answer 1

It is better to consider the composition of functions specifying range and domain. Let us work in the easy setting:

$$f:\mathbb R\rightarrow \mathbb R,~~z\mapsto f(z):=z, ~~(*)$$

and

$$y:\mathbb R\rightarrow \mathbb R,~~x\mapsto y(x):=\frac{1}{2}x.$$

When you write $f(y)=y$ in $1.$ you are just considering the function $f$ on any element $y$ (call it $y$, $z$, $x$ etc...it is the same: you are using $(*)$).

When you write $f(y)=x=2y$ in $2.$ notation is confusing, instead. I believe you should introduce the composition the composition $g:=f\circ y:\mathbb R\rightarrow\mathbb R$, where $$g(x)=f(y(x))=y(x)=\frac{1}{2}x.$$

Answer 2

I'll restrict this answer to the case where $f$ is a real function defined on the whole real line, ($f\colon \mathbb R\to \mathbb R$).

Since the domain of $f$ is $\Bbb R$, its input are real numbers. Therefore, the symbol $f(y)$ is meaningless if $y$ is not a real number.

In your example you set $f=\text{id}_\mathbb R$. Let $y$ be a real number. If it is the case that there exists $x\in \mathbb R$ such that $y=2x$, then $f(y)=f(2x)=2x$, by virtue of the definition of equality and of $f$.

If $y$ is a function (for simplication, let $\text{dom}(y)=\mathbb R$ and let it take only real values), then, as said before, $f(y)$ doesn't make sense. Something that makes sense is $f\circ y$ and, given $x\in \mathbb R$, $f(y(x))$ also makes sense because $y(x)\in \mathbb R$.

Now answering your question, the usual interpretation is that $y$ is a function and that you should look at $f(y)$ as $f\circ y$ (accompanied by a deep sigh), which is the convenient notation. Common terminology is saying that '$y$ is a function of $x$', which doesn't excuse the wrongness of $f(y)$.