Actual Meaning of a Curve

Question

Throughout my math education up until now (1st year of college) I had been told by all my math teachers that a curve that is drawn is "some letter parentheses $x$" (e.g. $f(x)$, $g(x)$, etc.). But one day I realized that $f(x)$ (let's just work with $f(x)$) is saying that the curve is made of only outputs when, actually, a curve is made of inputs and outputs. Then I asked on this site what the real name of a curve is and it's some letter (e.g. $f$, $g$, etc.), but why its just some letter is unclear to me. Anyways, staying on the main issue, If I have $f(x)$ and $f(-x)$, since $f$ is present in each case and since $f$ is a curve, the curves in each case should be the same. But they are different. Someone had told me $f(x)$ and $f(-x)$ are just different ways of looking at the same curve, but what does this mean? I have been trying to resolve this problem for 2 weeks.

Answer 1

I asked on this site what the real name of a curve is...

What you are calling a curve is what we call a function. The graph of a function is what I'm sure you are meaning by a curve. I should note, though, that a curve is NOT a function. They are two distinct entities, however they usually do relate to each other.

I had been told by all my math teachers that a curve that is drawn is "some letter parentheses x" (e.g. f(x), g(x), etc.)...a curve is made of inputs and outputs.

This is the notation we use to denote a function. If $f$ is a function, then we say $x$ is its input and $f(x)$ is its output. You can visualize this process by plotting $y=f(x)$ on the $x$-$y$ plane. For instance, if we have $f(1) = -1$, then we can go to the point associated with $x=1$ and $y=-1$ and draw a point there. We generally call this point $(1,-1)$ an ordered pair.

If I have $f(x)$ and $f(−x)$, since $f$ is present in each case and since $f$ is a curve, the curves in each case should be the same. But they are different. Someone had told me $f(x)$ and $f(−x)$ are just different ways of looking at the same curve, but what does this mean?

In general, $f(x) \neq f(-x)$, however there are exceptions. For example, if $g(x) = x^2$, then $$g(-x) = (-x)^2 = x^2 = g(x).$$ However, this isn't always true, for example if $f(x) = 3x+1$, then $$f(-x) = 3(-x) + 1=-3x+1\neq3x+1=f(x).$$ Remember, the notation $f(x)$ says that $x$ is our input and $f(x)$ is our output. This means that if $-x$ is our input, then $f(-x)$ is our output. To say that these are just different ways of looking at the same curve is sort of true. Informally, imagine taking the graph of $y=f(x)$ and flipping it (we usually say reflect) across the $y$-axis. Then you get $y=f(-x)$.

In the following figure, $f(x)=e^x$ is in red and $f(-x)=e^{-x}$ is in blue.

Consider reading the beginning of Understanding in Mathematics to understand functions.

Answer 2

The foundation of mathematics is generally taught using set theory, the crux of which being a collection of objects called a set: $A =\{1,2,3\}$. A point on a plane can be characterized as an ordered pair $(p,q)$, which is simply a set with two numbers with order. In other words $(1,2) \not = (2,1)$. This is a useful mathematical object.

We can characterize a bunch of points on a plane as a set of ordered pairs. People generally refer to sets of sets and collections. So check out this collection of ordered pairs: $A = \{(1,2),(5,3),(6,2)\}$. These are 3 points on a plane. If you think about it, a curve is nothing more than an infinity large collection of ordered pairs.

This was a revolutionary idea in mathematics that cropped up in the late 1800s, led by great minds such as Cantor. People never really thought things this way before. Before the 1800s, people generally thought of math as geometry: shapes, curves, etc. How would you define a curve? There was no definition! A curve was given, similar to how a set is given to us. Later on in Euler's time (about the late 1700s), mathematics transformed from being geometry-based to algebra-based. Instead of curves, you had functions. Instead of shapes, you had equations. Math taught in high school and some 300-level math classes tend to stick with this paradigm of thought, and for good reason. It's useful. However, it has flaws. What the heck is a curve? How is it different than a function? What does f vs. f(x) mean?

To know what a curve is, you have to know what a function is. And to know that, you need to know what binary relation is.

Take two sets $A,B$. A binary relation $F$ on $A,B$ is simply a set of ordered pairs where the first element is in the set $A$, and the second is in the set $B$. For instance, lets say $(1,3)$ is an element in $F$. We say that F maps 1 to 3. This will be useful later.

A function is a special type of binary relation. Specifically, it is a binary relation on some sets $A,B$ with the rule that every element in $A$ is mapped to a unique element in $B$. We call the left set the domain and the right set the codomain. So $F=\{(1,2),(5,6),(1,5)\}$ is not a function. It is however a binary relation on the sets $\{1,5\}$ and $\{2,5,6\}$. But not a function. This is because F maps 1 to 2 and maps 1 to 5. 1 is mapped to two elements. It isn't mapped to a unique element.

Take the binary relation $\{(1,2),(5,3),(2,2)\}$. This is a function. Sweet.

People like to use lowercase letters to represent functions, which mind you are simply collections of ordered pairs with that function rule that every element in the domain has a uniquely mapped element in the codomain.

Let $f$ be a function that maps $\mathbb{R}$ to $\mathbb{R}$. The notation we use is $f:\mathbb{R} \rightarrow \mathbb{R}$. We define $f$ as $f(x)=x^2$. So $f(x)$ is a notation used to denote whatever the number $x$ is mapped to. What should $f(x)$ and $f(-x)$ be any different? Well $x$ is just an arbitrary number. Therefore $-x$ is the negation of an arbitrary number, which is itself an arbitrary number. From a logical standpoint, they are identical.

So yes, $f$ is a curve, but $f(x)$ isn't. Though some people use the notation interchangeably and that's fine. Notation is all about exchanging information, so if it's doing that then good.

Note that $f$ isn't technically a curve. I can define some pretty weird functions that are discontinuous. A curve has to be continuous, but that's a story for another time.