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I've learned a basic definition for function in calculus and I noticed that in:

$$f(x)=[formula]$$

We take one number in the $LHS$ (which doesn't involve calculations, we just pick a number) and then we use this number in the $LHS$ of the function, which will require calculations. For example:

$$f(x)=x^2$$

We choose a number for $x$, say $2$ and then we use it in the other side and then perform the required calculations. I just had the idea of doing a function that involves calculations on both sides and plot their results, I'm using Mathematica for this and doing the following code:

ListLinePlot[Table[{Sin[x], Cos[x]^10}, {x, 0, 10, 0.01}]]

Yields this plot:

enter image description here


Using $f(x)=x^2$, I know that it's plot is going to be all the pairs $\{x,x^2\}$, the above function is going to be $\{\sin x,\cos x^2\}$.


This code:

ListLinePlot[Table[{Sin[x], Cos[x]^11}, {x, 0, 10, 0.01}]]

Yields this:

enter image description here

This code:

ListLinePlot[Table[{Sin[x], Cos[x]*Sin[x]}, {x, 0, 10, 0.01}]]

Yields this:

enter image description here

Which I guess that is the Lemniscate of Bernoulli (already associated it's form with it's name, but I know nothing about it's properties). It is surprising because I was kinda used to the vertical (and sometimes) horizontal tests which fail in these functions. I found this by pure accident and I have no idea of what I've done, I just noticed the similarity with polar plots (which I still don't know much what they really are). So, is there a name for this function or these are just accidentally found polar plots? What I've done here?


Edit: One thing I almost forgot, I've tried with exponential functions:

ListLinePlot[Table[{x^8, x^3}, {x, 0, 10, 0.001}]]

enter image description here

ListLinePlot[Table[{x^3, x^8}, {x, 0, 10, 0.001}]]

enter image description here

I noticed that in the case $\{x^a,x^b\}$, if $a>b$, the curve bends to the top, if $a<b$, the curve bends to the bottom, if $a=b$, the function seems to be similar to a linear function (which is a trivial conclusion). I'm trying to think if this has some deeper meaning.

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A function such as $f(x) = x^2$ is called a real-valued function of one real variable. This means that the function takes one real value as input, $x$, and returns one real value as output, $f(x)$. We write $$f: \mathbb{R} \to \mathbb{R}$$ to indicate this. The usual way to plot such functions is to plot all the pairs $(x, f(x))$ in a coordinate system.

A function having the pairs $(\sin t, \cos t^2)$, $t \in \mathbb{R}$, as its plot, takes one real variable $t$ but returns two real numbers $\sin t$ and $\cos t^2$. We can express this by saying that it return an ordered pair $(g(t), h(t))$ for each value of $t$ (here $g(t) = \sin t$ and $h(t) = \cos t^2$). Thus, if we denote by $\mathbb{R}^2$ the set of pairs of real numbers, we have a function $$F: \mathbb{R} \to \mathbb{R}^2$$ Given by $F(t) = (\sin t, \cos t^2)$.

In general we can have functions of the form $G: \mathbb{R}^n \to \mathbb{R}^m$, which take $n$-tuples of real numbers as input and returns $m$-tuples of real numbers as output.

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