I don't understand why we represent functions $f:I \subseteq \Bbb R \to \Bbb R^2$ the way we do.

Question

I don't understand why we represent functions $f:I \subseteq \Bbb R \to \Bbb R^2$ the way we do: doing an analogy with how we represent functions from $\Bbb R$ to $\Bbb R$ or from $\Bbb R^2 \to \Bbb R$, my impression would be to represent functions like $f$ in the format: $$(t,f_1(t),f_2(t))$$

But instead we grab all the points $(f_1(t),f_2(t))$, that is, the image of $f$ and we plot that in $\Bbb R^2$, and we call $f$ a parametrization.

As an example of this, let $f(t)=(\cos t, \sin t)$. If we represented this by plotting the points $(t,\cos t, \sin t)$ we'd get a helix. But instead we 'splash' that helix into the wall and we get the parametrization of a circle.

Answer 1

We do not "represent" functions $f:I \subseteq \Bbb R \to \Bbb R^2$ as $(f_1(t),f_2(t))$.

The functions $f:I \subseteq \Bbb R \to \Bbb R^2$ are represented as $(t,f_1(t),f_2(t))$ (where $t$ varies on some subset of real line) just like other kinds of functions.

$(f_1(t),f_2(t))$ (where $t$ varies on some subset of real line) is the set which is called the image of function $f$.

Image of a function is an important concept, but "image of a function" ($(f_1(t),f_2(t))$ over varying $t$) and "graph of a function" ($(t,f_1(t),f_2(t))$ over varying $t$) are two different concepts. You should just understand that both these two: a. exist; b. are important; c. are different.

Answer 2

Because it's a useful simplification sometimes. Explicit time dependence is really scary, actually. Imagine I'm watching a system that shouldn't depend explicitly on time (earth's gravitational field shouldn't fluctuate based on time, for example). But I'm watching a ball travel and its position does depend on time. If the ball is at position $(x,y)$ at time $t_1$ and again at time $t_2$ I want to say that the ball is orbiting.

But if I imagine the ball's coordinates as having a time component, the it will never hit the same point twice. That is $(t_1,x,y) \neq (t_2,x,y)$. So the ball would never really be in orbit.

This idea is strongly reflected in the field of differential equations, where systems that do not explicitly depend on time are much better understand than those that do. Things like poincare-bendixon theorem (which are awesome) depend heavily on the notion of orbits. "If a ball hits the same point twice its path must be periodic" is a critical lemma for the theorem.