Functions

This question is conceptual. All throughout my young'un days, I was told that "functions must be one-to-one". All curves that do not pass the "vertical line test" are not functions. All curves that do not pass the "horizontal line test" do not have inverses.

Why do functions have to be "one to one"? I don't understand the purpose of "not having two outputs for a single input". If I want to visualize the plot of f(x)=sin(x), I would plot all possible points in the domain t=[-∞,∞] for (x,y)=(t,sin(t)). I can also visualize reflections over the line y=x (and equivalently x=y), by plotting all possible points in the domain t=[-∞,∞] by d(t)=(sin(t),t). Yet supposedly, d(t)=(sin(t),t) is not a function (and consequently not an "inverse") because it "doubles-back on itself".

x=sin(y) cannot possibly be a function; no... no way.

It's like that because it's the way it's defined. Most definitions for functions wouldn't work if you consider they can not pass the "vertical line test", how do you define limits, integrals, derivatives, increasing/decreasing, etc?

I'm asking for the reason behind the definition, not the definition itself.

Limit: (x,y) = (limit of x as t approaches a, limit of y as t approaches a)
Derivative: dy/dx = (dy/dt)/(dx/dt)
Integral: d⁻¹y/dx⁻¹ = (d⁻¹y/dt⁻¹)/(d⁻¹x/dt⁻¹)
Increasing: dy/dx = (dy/dt)/(dx/dt) > 0
Decreasing: dy/dx = (dy/dt)/(dx/dt) < 0

And yes, I'm aware that x(t) and y(t) are one to one functions of t, but many curves are like not one-to-one but can be represented as a composition of one-to-one functions. Yet, they are not considered "functions". For example, the unit circle on the xy plane would fail to be a function, yet it has well-known properties and a well-defined parametric representation.

 Off topic

If anyone knows:

Why is Γ(z)=(n-1)! considered the "best" representation of the factorial function, even though it's divergent at every negative integer? What does it mean to have a negative factorial anyways? Because of fractional calculus, should I be under the impression that (-n)!=1/n!? Why does Γ(z) suggest otherwise. Anyways, I'm just curious because I saw the other representations on this page. I don't have actual values for Hadamard's Gamma (and I couldn't find any), but from what it looks like, (-n)!=1/n! on Hadamard's Gamma.

First, as for how: A function is a relation between each element in the domain and its corresponding element in the range. Scenarios where it's not one-to-one:

- Zero-to-something: If there's nothing to find a corresponding soul mate for, end of story. The relation can not be defined. (E.g: f(x) = x + 2, for strictly x ∈ [0, 2] is not defined at x = -1)

- Something-to-zero: Same thing, though I believe we tend to remove the value from the domain if there's no meaningful corresponding value in the range that can be assigned to it. (E.g: f(x) = 1/x, x ∈ (-∞, +∞ is not defined at x = 0 because 1/0 is not a meaningful mathematical quantity, thus we remove 0 from the domain, even though it spans from -∞ to +∞

- One-to-many: I just don't think it's useful to be defined this way. Take, for example, the following function:

Code:

f(x) =
          1 + x, if x ∈ (0, 1)
          x - 2, if x ∈ (-2, 0)
          0, if x = 0
          36x + 49, if x != 0

What does it even mean to have many outputs for one input? There will have to be a second factor that decides the output, making things all the more complicated. I think function is among the basic building blocks for Mathematics, if you want to build more complicated relations (like one-to-many input-output relations), you can tweak the basic functions to have what you want (E.g: Parametric equations). Even if you allow for a "function" to have multiple output, there will still exist the concept of 1-to-1 relations (within each of those multiple outputs), which is more fundamental than your definition of a function. What to call it?

- Many-to-one: Perfectly fine in its own scope. Each soul searcher will find someone they can relate to. However, as you've already mentioned, the reverse wouldn't be true.
I don't think a function excludes this case, as there is no indication of any requirement for reversibility for a relation to be defined as a function.

And yes, I'm aware that x(t) and y(t) are one to one functions of t, but many curves are like that and not considered "functions". For example, the unit circle on the xy plane would fail to be a function, yet it has well-known properties and a well-defined parametric representation.

*coughs on a discussion of parametric and polar functions*

In parametric mode

x=cos(t)
y=sin(t)

and you got yourself a function!

Why do functions have to be "one to one"? I don't understand the purpose of "not having two outputs for a single input".

There are several reasons for it: First of all, mathematics is mostly about precise definitions. What would you do if you had a function f which has the mapping 2 → 3 and 2 → 4?
Is f(2) = 3 or 4? If f(2) = 3, wouldn't that mean that 3 = 4 (because f(2) = f(2) and f(2) = 3, f(2) = 4)? If functions were a broader term and you had the possibility to have several mappings for one x to multiple ys, you couldn't use normal operations nor equality. And without equality, things look pretty grim in mathematics.

Basically, having many-to-one or many-to-many functions would require a pretty drastic redefinition of mathematics and would make it all more complicated.

If anyone knows:

Why is Γ(z)=(n-1)! considered the "best" representation of the factorial function, even though it's divergent at every negative integer? What does it mean to have a negative factorial anyways? Because of fractional calculus, should I be under the impression that (-n)!=1/n!? Why does Γ(z) suggest otherwise. Anyways, I'm just curious because I saw the other representations on this page. I don't have actual values for Hadamard's Gamma (and I couldn't find any), but from what it looks like, (-n)!=1/n! on Hadamard's Gamma.

Well, it's more generally defined as



It is not defined for non-negative numbers. Hadamard's Gamma does not have the same properties as Euler's Gamma. Euler's Gamma is the only function which is log-convex, has f(x) = 1 and f(x + 1) = xf(x). I assume you know of everything but the log-convex-part, so let's just take a short proof on that:

By the definition of the Euler-gamma, Γ(X) is positive for positive values of x. If you then let x, y > 0 and 0 ≤ λ ≤ 1, then



Thus, it is log-convex. And it's quite important, for example, in computer science, that it is log-convex. If it wasn't, one couldn't (in some cases) show lower and upper boundaries on the time complexity of a computer program/algorithm.

Noah

Apart than computer algorithms, what's so special about log-convex functions? Is it so important that it's better than being "correct"?

I know that when Euler's gamma was made, the function wasn't meant to be extended into negative numbers, but people ended up doing so anyways. Now I'm wondering why people didn't consider defining an interpolation function for (-n)!.

If you look at the definition of factorial in terms of the previous value, it's
fact(x) = x*fact(x-1)
fact(0) = 1
for x in non-negative integers.

If you try to extend that to include negative integers, the obvious method is to invert that function so you can go towards the negatives:
fact(x-1) = fact(x)/x
fact(-1) = 1/0
fact(-2) = (1/0)/-1 = (-1/0)

Either you redefine factorials some other more complicated way, or you deal with it being undefined at negative integers. Might as well keep it simple. It still works for real numbers, anyway.

Apart than computer algorithms, what's so special about log-convex functions? Is it so important that it's better than being "correct"?

I know that when Euler's gamma was made, the function wasn't meant to be extended into negative numbers, but people ended up doing so anyways. Now I'm wondering why people didn't consider defining an interpolation function for (-n)!.

I would like to avoid discussing whether it is better or not - or more "correct" for that matter - but it is worth noting that it is the first function of its kind, it is very simple compared to other functions (and thus is easier to compute, too), and have this log-convex property. I also think, but I am not sure, that gamma applications for negative values are sparse, and we have stayed with this function because it works for most applications.

As Euler's Gamma function is log-convex, that also means it is more "smooth": It does not make those bumps and jumps as Hadamard's Gamma does. It is quite important that it works like that for e.g. the gamma distribution, whenever you use it with non-integer alpha values and want the correct probability. Whether that has anything to do with the usage of Euler's Gamma as the "standard" gamma function, I do not know, but I am quite sure it does not.

Noah

A function is, by definition, one-to-one because we already have a term for a relation which is not necessarily one-to-one.

What you're asking is akin to something like, why does a square have to have equal sides? Rectangles don't have equal sides and they're perfectly good geometric figures.

It's useful to have a name for a one-to-one relation. Many ideas can be expressed in terms of functions. At any given location, the temperature at that location is a function of time. You can't have two temperatures at the same time.

Edit: "One-to-one" actually refers to a function that is invertible, that is, only one input for every output. Otherwise, it would be redundant terminology.

http://en.wikipedia.org/wiki/Injective_function

Yeah, functions can be one-to-one but they can also be many-to-one - for example, f(x) = 3 maps all values to one. It's not invertible, but it is a fine function.