Coordinates versus vectors in functions

Question

Since I have had a course in linear algebra I have the following question:

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^2$

How should I interpret this function?

1) As all vectors $(x,y) \in \mathbb{R}^2$ that satisfy $y = x^2$

2) Or as all coordinates $(x,y)$ seen from a certain fixed basis that satisfy $y = x^2$

I think it depends a bit on the context, but I would like to know how I have to think about such functions. Maybe this is a bad example, as we can just look at this function as a correlation between x and y without thinking about it as functions. But what if we consider an isometry or a function that we can geometrically interpret.

Answer 1

Strictly speaking, functions from $A$ to $B$ are a special case of relations, and these are defines simply as subsets of $A\times B$.

So, a function $f:\mathbb R\to\mathbb R$, defined as $f(x)=x^2$ is, by definition, a set of ordered pairs $(x, y)$ such that $y=x^2$.

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The question of coordinates vs vectors isn't really a question, since it simply depends on what structure you put on $\mathbb R^2$. If you look at $\mathbb R^2$ as a vector space, then its elements are vectors (but they are still, in essence, ordered pairs). If you look at it as a coordinate system, then its elements are coordinates.

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TL;DR: the function is a set of ordered pairs. These pairs can be intepreted as either vectors or coordinates (just like I am a human, but also a European and an Earthling).

Answer 2

You should interpret an $f : \mathbb{R} \to \mathbb{R}$ as an object that, when plugged into the evaluation operation along with an $x : \mathbb{R}$, produces a result $f(x) : \mathbb{R}$. Furthermore, for $x : \mathbb{R}$, you have $f(x) = x^2$.

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You can talk about things along the lines of what you suggest (and often, people talk about functions when they really have such things in mind). I'm not up to trying to introduce them in this post, but one direction for searching is the keyword "scalar field". Another direction might be to look into formal logic.