With having a deeper understanding of functions, I am now revisiting using functions in topics such as physics and have a question regarding the notation “$f(x)=y$“ for a function $f: X \to Y$ (Where $x \in X$ and $y \in Y$).
When we say “$f$ depends on $x$” are we really saying “the value of $f$ depends on $x$“? With that said, since the value of $f$ is understood to be $y$, doesn’t that mean $y$ depends on $x$?
Suppose we have some set of points $D \subseteq X$ and $R \subseteq Y$. In general, the coordinates $(x,y)$ of a point $P \in D \times R$ will be completely independent of one another, as we assume them to be independent variables. But, in some special cases they will be related by some relation, like $f(x,y)=c$ or $f(x,y)<c$ and so on. $x^2 + y^2=r^2$ is an example. However in an even more special case, the value of one of the coordinates, say $y$, will completely depend on the other coordinate, $x$, via some function $f$. Formally speaking, $\forall (x,y) \in D\times R$, $y=f(x)$. In this way, we can reduce all problems involving two variables, $x$ and $y$ to problems about $x$ and $f(x)$, which is a problem of one variable. Remember that $f$ is a function, a list of rules to follow, whereas $f(x) \in R$ and is a value.