# What is Function: Definition and 1000 Discussions

In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. It is customarily denoted by letters such as f, g and h.If the function is called f, this relation is denoted by y = f (x) (which reads "f of x"), where the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (e.g., f is a function of the variable x).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means of illustrating the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.

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1. ### Injective function

I operated by placing ##S## and ##T## to two singlets belonging to ##X## and therefore established that for ##T, S \in X##, therefore ##f (T) = f (S) \implies S = T##, consequentially: $$f (T \cap S ) = f (T \cap T) = f (T) \cap f (T) = f (T) \cap f (S)$$. I would like to know if my procedure is...
2. ### I Question on Lambert W function

In the following I ask WA to solve the given equation and it produces a solution using the Lambert W function. I thought : $$W(x*e^x) = x$$ but here it seems $$W_n \left(\frac{-MT}{P}*e^{\frac{-MT}{P}}\right) \neq \frac{-MT}{P}$$ Is there a difference between ##W(x)## and ##W_n(x)## ?
3. ### B Looking for a specific periodic function

Is there a function that outputs a 1 when the input is a multiple of a number of your choice and 0 if otherwise. The input is also restricted to natural numbers. The only thing I can come up with is something of the form: f(x) = [sin(ax)+1]/2 but this does not output a 0 when I want it.
4. ### POTW A Function in the Continuous Hölder Class

Let ##0 < \alpha < 1##. Find a necessary and sufficient condition for the function ##f : [0,1] \to \mathbb{R}##, ##f(x) = \sqrt{x}##, to belong to the class ##C^{0,\alpha}([0,1])##.
5. ### Partial Derivative Simplification

Hi there! I would like to know if the following simplification is correct or not: Let A be a function of x, y, and z $$\frac{\partial^2A}{\partial x^2}+\frac{\partial^2A}{\partial y^2}$$ $$=\ \frac{\partial^2A\partial y^2+\partial^2A\partial x^2}{\partial x^2\partial y^2}$$...
6. ### I Understanding Hessian for multidimensional function

Hello everybody, I have a question regarding this visualization of a multidimensional function. Given f(u, v) = e^{−cu} sin(u) sin(v). Im confused why the maximas/minimas have half positive Trace and half negative Trace. I thought because its maxima it only has to be negative. 3D vis 2D...
7. ### Find the derivative of the given function

Let's see how messy it gets... ##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##...
8. ### Show that the given function is decreasing

As a follow up for : https://www.physicsforums.com/threads/let-k-n-show-that-there-is-i-n-s-t-1-1-k-i-1-2-k-i-1-4.1054669/ show that ## \alpha\left(k\right)\ :=\ \left(1-\tfrac{1}{k}\right)^{\ln\left(2\right)k}-\left(1-\tfrac{2}{k}\right)^{\ln\left(2\right)k} ## is decreasing for ##...
9. ### For this Partial Derivative -- Why are different results obtained?

Given a function F(x,y)=A*x*x*y, calculate dF(x,y)/d(1/x), to calculate this derivative I make a change of variable, let u=1/x, then the function becomes F(u,y)=A*(1/u*u)*y, calculating the derivative with respect to u, we have dF/du=-2*A*y*(1/(u*u *u)) replacing we have dF/d(1/x)=-2*A*x*x*x*y...
10. ### I What are the Zeta Function and the Riemann Hypothesis?

What is the zeta function and the Riemann hypothesis.
11. ### I Fermi energy for a Fermion gas with a multiplicity function ##g_n##

I ran across the following problem : Statement: Consider a gas of ## N ## fermions and suppose that each energy level ## \varepsilon_n## has a multiplicity of ## g_n = (n+1)^2 ##. What is the Fermi energy and the average energy of this gas when ## N \rightarrow \infty## ? My attempt: The...
12. ### Vector is a function of its position or not?

At first I thought that this force vector ## \vec F = 3 \hat x + 2 \hat y ## is a function of ## x ## and ## y ##, which is to say that its magnitude and direction vary with the x and y positions, but this is not so, right? It's just a force with a constant magnitude and direction. And I can...
13. ### Distance as a function of time for two falling stones

I am aware that this question is very simple and basic. Using ##y(t)=y_0+v_{0,y}t-\frac {1}{2}gt^2## we can find distance as a function of time: ##|y_1-y_2|=|y_0+v_{0,y}t|=-y_0- v_{0,y}t## I assumed the downward direction to be negative. So as I wrote ##D(t)=-y_0- v_{0,y}t##. It tells that the...

38. ### Expressing Feynman Green's function as a 4-momentum integral

I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do...
39. ### B How do I invert this exponential function?

Preface: I have not done serious math in years. Today I tried to do something fancy for a game mechanic I'm designing. I've got an item with a variable power level. It uses x amount of ammo to produce f(x) amount of kaboom. Initially it was linear, e.g. fL(x) = x, but I didn't like the scaling...
40. ### I Best fit to an oscillating function

Hello! I have a plot of a function, obtained numerically, that looks like the red curve in the attached figure. It is hard to tell, but if you zoom in enough, inside the red shaded area you actually have oscillations at a very high frequency, ##\omega_0##. On top of that you have some sort of...
41. ### I Limit as a function, not a value

Is it possible for a limit of a range of functions to return a function? Example: f(z)= limit (as p approaches 0) (xp-1)/p.
42. ### Finding the domain of a composite function

For this problem, The solution is, However, I tried solving this problem by using the definition of composite function ##f(g(x)) = f(\frac{4}{3x -2}) = \frac{5}{\frac{4}{3x - 2} - 1} = \frac{5}{\frac{6 - 3x}{3x - 2}} = \frac {15x - 10}{6 - 3x}## which only gives a domain ##x ≠ 2##. Would some...
43. ### I Integration of Bessel function products (J_1(x)^2/xdx)

Hello, While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral ## \int_0^\infty J_1(x)^2\frac{dx}{x}=1/2 ## I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations...
44. ### I Geometry of series terms of the Riemann Zeta Function

This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i## The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...

50. ### Trying to reconcile function composition problems with sets & formulas

I know how to solve each of those problems. For the set one, I look at the output of the S and try to match it with the input of T and then take the pair (input_of_S, output_of_T), and I do that for each pair. As for the formula one, I just plug in x = g(y). My confusion lies in trying to...