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.

View More On Wikipedia.org
Recent contents Top users
  1. ttpp1124

    Determine the equation of the tangent line to the function given

  2. abivz

    I Obtaining the Dirac function from field operator commutation

    Hi everyone, I'm new to PF and this is my second post, I'm taking a QFT course this semester and my teacher asked us to obtain: $$[\Phi(x,t), \dot{\Phi}(y,t) = iZ\delta^3(x-y)]$$ We're using the Otto Nachtman: Elementary Particle Physics but I've seen other books use this notation: $$[\Phi(x,t)...
  3. M

    A Nowhere diffferentable continuous function

    Weierstrass function is the classic example of a continuous function which is nowhere differentiable. What happens when a function is monotone? My guess that it cannot be nowhere differentiable. It seems to me the reverse is true - it is differentiable almost everywhere. Any light on the...
  4. tworitdash

    A Spatial Fourier Transform: Bessel x Sinusoidal

    I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m + 1)...
  5. G

    MHB Minimize a function: Find value of x that result in lowest value of formula

    Hi, I have this formula, What I want is to find the value of "x" (without trying all possibilities) so that the result of the formula will be the lowest possible value under the constraint when x !=0, and x<n. Here, values of A,B,C, Q, R,n are already known and fixed...
  6. maistral

    Reaction kinetics + Gillespie algorithm: Propensity function?

    I'm trying to simulate a simple series reaction stochastically using Gillespie's algorithm. I found this file: What is this 'propensity function'? Say for example I have the simple reactions: A --(k1)--> R R--(k2)--> S are these 'propensity functions' the rates (a wild guess)? I mean; α1 =...
  7. E

    Function for the movement of a charged particle in a B field

    The movement in the z-direction is easy to solve for, as it's only affected by the gravitational force. However, if there's a magnetic field pointing down along the z-axis, the particle is going to be accelerated along the y-axis (F=q*v *B). The force is always going to be perpendicular to the...
  8. R

    I Inverse Laplace transform of a rational function

    I struggle to find an appropriate inverse Laplace transform of the following $$F(p)= 2^n a^n \frac{p^{n-1}}{(p+a)^{2n}}, \quad a>0.$$ WolframAlpha gives as an answer $$f(t)= 2^n a^n t^n \frac{_1F_1 (2n;n+1;-at)}{\Gamma(n+1)}, \quad (_1F_1 - \text{confluent hypergeometric function})$$ which...
  9. A

    Comp Sci Why Does Using an Uninitialized Pointer Cause a Segmentation Fault in Recursion?

    double foo(int arr[], double *ave, int index){ double *s; *s=*ave; // calculation// return(foo (arr,ave,index)); // other calculation// } I want to keep the ave value during the recursion, because after ave is calculated, I will do another calculation is recursively in this...
  10. P

    MATLAB How to calculate Bessel function of order zero?

    Hello everyone. I try to plot a figure from a journal article. I gave the equations in the inserted image. I wrote the script given below for that. I expect to obtain a plot like the one given on the left but I end up with something totally different. So, the values of ##I_{0}## and ##I_{1}##...
  11. Physics Dad

    How Accurate is the Initial Mass Function in Predicting Stellar Distributions?

    Assumptions: 1) The minimum stellar mass in this cluster is 0.1M⊙ 2) The maximum stellar mass in this cluster is 150⊙ First calculate the local stellar density constant (ξ0) for this cluster using eq 1: Having rearranged this equation and using the limits of the minimum and maximum stellar...
  12. The black vegetable

    I Vertex function, quantum action

    I am looking at Srednicki ch 64 , how does equation 64.1 follow from 64.3 as stated. Explicitly in QED how does ## u_{s'}(p')V^{u}(p',p)u_{s}(p)=e\bar{u'}(F_{1}(q^{2})\gamma ^{u}-\frac{i}{m}F_{2}(q^{2})S^{uv}q_{v})u ## follow from the quantum action ## \Gamma =\int d^{4}x(eF_{1}\bar{\varphi...
  13. CaptainAmerica17

    I Is f(x) an Injective Function? Understanding Proof and Notation

    I typed this up in Overleaf using MathJax. I'm self-studying so I just want to make sure I'm understanding each concept. For clarification, the notation f^{-1}(x) is referring to the inverse image of the function. I think everything else is pretty straight-forward from how I've written it. Thank...
  14. M

    MHB Weierstrass Function: Continuous and Bounded on $\mathbb{R}$

    Hey! :o I am looking at the following example of a continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ that is not differentiable at any $x\in \mathbb{R}$. For $x\in [-1,1]$ we define $\phi (x)=|x|$ and then we extend $\phi$ to the whole $\mathbb{R}$ such that $\phi (x+2)=\phi (x)$...
  15. pellman

    I Behavior of a function for large x?

    I have a problem asking to show that a certain function approaches a quadratic for large values of the variable. And I realize now that this is a skill with which I am totally unfamiliar. Can't use a Taylor series in y= 1/x because the value at y=0 is infinite. Would appreciate a recommended...
  16. Cesca Roma

    I Discriminant function analysis - stepwise or otherwise?

    I’m using discriminant function analysis to determine the potential accuracy of several biometric measurements being used in conjunction for binary classification purposes for my BSc Biomed research project. Overall I've only got 110 data points so it's a stretch but hey, that's anatomy! What...
  17. agnimusayoti

    Limit of a function as n approaches infinity

    If there is no ##(-1)^2## factor, I can find the limit. But, now I have no idea how to find limit for the ##(-1)^\infty##. I thought ##(-1)^\infty## is an indeterminate form. So, how to modify this? Thanks!
  18. Physics lover

    No. of solutions of an equation involving a defined function

    Here is a pic of question My attempt-: I defined functions f(x-1) and f(x+1) using f(x).After defining them,I substituted their values in the equation f(x-1)+f(x+1)=sinA. For different ranges of x,I got different equations. For 1<x<2,I got 1-x=sinA. But now I am confused.For each different...
  19. N

    Mathematica How to define such a simple function in Mathematica

    A very simple question but I can't find an answer. I have an expression which depends on two integers, n,d. Now, I want this expression to be a) 1 when d=n=0, b) some expression (that I won't write here) when both d and n are >0 c) zero when wither d or n negative. At first I defined the...
  20. dykuma

    Evaluating an integral of an exponential function

    the integral is: and according to mathematica, it should evaluate to be: . So it looks like some sort of Gaussian integral, but I'm not sure how to get there. I tried turning the cos function into an exponential as well: however, I don't think this helps the issue much.
  21. anemone

    MHB Roots of a Polynomial Function A²+B²+18C>0

    If a polynomial $P(x)=x^3+Ax^2+Bx+C$ has three real roots at least two of which are distinct, prove that $A^2+B^2+18C>0$.
  22. Zouatine

    What units for the wave function of a string?

    hello , hope all of you are doing well , i have question about the unit of the function of waves of string fixed in both boundary , the function of waves is function of two variables x and t , so it's function describe the displacement in function of place and time , Ψ(x,t)=φ(x)*sin(ωt+α)...
  23. C

    Proving a function is injective

    Hello, Let f: ]1, +inf[ → ]0, +inf[ be defined by f(x)=x^2 +2x +1. I am trying to prove f is injective. Let a,b be in ]1, +inf[ and suppose f(a) = f(b). Then, a^2 + 2a + 1 = b^2 + 2b + 1. How do I solve this equation such that I end up with a = b? Solution: (a + 1) ^2 = (b + 1)^2...
  24. evinda

    MHB Interval with Dirac function in a finite interval

    Hello! (Wave) I want to calculate the integral $\int_{-1}^2\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt$. I have done the following so far: $$\int_{-\infty}^{+\infty}\sin \left (\pi (t-1)\right )\delta (-t+1)\, dt=\int_{-\infty}^1\sin \left (\pi (t-1)\right )\delta (-t+1)\...
  25. Adesh

    Why the existence of the potential function ##U## is not sufficient?

    In Sommerfeld’s Lectures on Theoretical Physics, Vol II, Chapter 2, Section 6, Page 43 we derive an expression for the equilibrium of liquids as $$ grad ~p = \mathbf F$$ Where ##p## is the pressure and ##F## is the exertnal force. Then he writes, [ The equation above ]includes a very remarkable...
  26. A

    Displacement as a discrete function of time

    Given initial displacement ##X_0## and displacement at any time ##t## as ##x##. Where ##x(t)=f_t(X_0)## where the functional dependence of ##x## upon ##X_0## changes with time. For exm ##X_0=2## and ##x(t_1)=X^2_0=4,x(t_2)=X^2_0+1=5,x(t_3)=X_0^3+3=11...##and so on. From this, is there any method...
  27. S

    I Splitting of a one-particle wave function

    Hello all, I am a newcomer here. Not a physicist, just an enthusiast. ;) I was thinking whether it is possible to separate a one-particle wave function into two, "completely disjoint" parts. The following thought experiment explains better what I am thinking about. Let us suppose, that there...
  28. caffeinemachine

    MHB Maximum value a function satisfying a differential equation can achieve.

    Let $f:\mathbb R\to \mathbb R$ be a twice-differentiable function such that $f(x)+f^{\prime\prime}(x)=-x|\sin(x)|f'(x)$ for $x\geq 0$. Assume that $f(0)=-3$ and $f'(0)=4$. Then what is the maximum value that $f$ achieves on the positive real line? a) 4 b) 3 c) 5 d) Maximum value does not exist...
  29. karush

    MHB 3.4.6 limit of a power function

    Ok all I did was DesmosNot real sure how to take limit
  30. V

    Minimizing a function in python

    The function is f(x)=x5-12x3+7x2+2x+7. I found the minimum of the function and compared the value to a calculator and it seemed okay. But I am confused as to how to incorporate the interval into my code. Has my code already sufficiently answered the question? from scipy import optimize...
  31. G

    MHB What is the Limit of an Exponential Function?

    Hello everyone, can anybody solve this limit? This is really tough one for me, thank you in advance.
  32. S

    I Derivative of a complex function along different directions

    Below are plots of the function ##e^{0.25(x-3)^{-2}} - 0.87 e^{(x-3.5)^{-2}}## The first plot is for real values. It has a minimum at the red dot. The second plot has in its argument the same real part as the red dot, but has the imaginary part changing from -0.3 to 0.3. It shows the resulting...
  33. CrosisBH

    Computing the wave function of a square potential

    The book's procedure for the "shooting method" The point of this program is to compute a wave function and to try and home in on the ground eigenvalue energy, which i should expect pi^2 / 8 = 1.2337... This is my program (written in python) import matplotlib.pyplot as plt import numpy as...
  34. S

    I Rewriting a piecewise function using step functions

    Suppose we have a piecewise function f(t) = exp(c*t) when 0 <= t < 2 and f(t) = 0 when t >= 2. Can the above be rewritten as f(t)= exp(at)*[H(t-0) - H(t-2)], H is a heaviside function.
  35. Tony Hau

    Potential energy as a function of the square of this angle

    The problem of my question is the b part below: I know that the potential energy is just the gravitational potential energy, which is mgh(𝜃) = mg[(R+b/2)cos𝜃 +R𝜃sin𝜃], derived from the geometry. The equilibrium point is at 𝜃=0 and the system is a stable equilibrium for R>b/2. However, I have no...
  36. C

    A Evaluation of an improper integral leading to a delta function

    Hi, I have pasted two improper integrals. The text has evaluated these integrals and come up with answers. I wanted to know how these integrals have been evaluated and what is the process to do so. Integral 1 Now the 1st integral is again integrated Now the text accompanying the integration...
  37. snatchingthepi

    Partition function from the density of states

    I'm given the following density of states $$ \Omega(E) = \delta(E) + N\delta(E-\Delta) + \theta(E-\Delta)\left(\frac{1}{\Delta}\right)\left(\frac{E}{N\Delta}\right)^N $$ where $ \Delta $ is a positive constant. From here I have to "calculate the canonical partition function as a function of $$...
  38. Zack K

    Infinite Square Well with polynomial wave function

    Some questions: Why is this even a valid wave function? I thought that a wave function had to approach zero as x goes to +/- infinity in all of space. Unless all of space just means the bounds of the square well. Since we have no complex components. I am guessing that the ##\psi *=\psi##. If...
  39. P

    Partition Function for Spin-1 One Dimensional Ising Model

    $$H=-J\sum_{i=1}^{N-1}\sigma_i\sigma_{i+1}$$ There is no external magnetic field, so the Hamiltonian is different than normal, and the spins $\sigma_i$ can be -1, 0, or 1. The boundary conditions are non-periodic (the chain just ends with the Nth spin) $$Z=e^{-\beta H}$$...
  40. S

    B What is the speed of a photon traveling along the sine function?

    On the image you can see a photon starting at point A at t=0. The photons travels along the sine function and arrives point C. I knot that this takes T=λ/c. But this is the time for a object traveling directly from the origin to point C and not along the sine wave! If the photon travels...
  41. filip97

    Carnot function -- How can I prove f(t2,t1)=f(t2−t1,0)

    If we have that quotient of heats ##Q_2/Q_1=f(t_2,t_1)##, where ##t_1,t_2## are emirical temperatures. Is this function satisfies : ##f(t_2,t_1)=f(t_2-t_1,0)## I try prove it with Taylor series of two variables, but i can't prove anything.
  42. nuclearfireball_42

    Can I determine the phase angle of this equation by using the sin function?

    I've got the answer for (a). It's k = 0.78 N/m. I'm having problems with (b). I know that the equation of displacement in this case should either be : x(t) = Asin(ωt + φ) or x(t) = Acos(ωt - φ) where A = amplitudeFrom what I understand, both the equation above should give the same result as...
  43. O

    How to prove this statement about the derivative of a function

    My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...
  44. karush

    MHB Mean Value Theorem: Showing Change in a Function is Bounded

    Ok Just have trouble getting this without a function..
  45. hilbert2

    A Can Bloch Waves Reveal Periodic Potentials in Quantum Mechanics?

    I was thinking about a problem I had considered a long time ago in some thread, finding an example of a wave function ##\displaystyle \psi (x) =e^{iax}\phi (x)## with ##\displaystyle\phi (x)## being periodic with period ##\displaystyle L## and the corresponding Schrödinger equation...
  46. zilex191

    Simple harmonic motion equations as a function of time

    I conducted a mass-sprig experiment to see how stiffness of a spring and mass affect the frequency of oscillation. In addition to this to this i have to plot a graph to show displacement,velocity and acceleration of the mass as a function of time.From my research online For the displacement as...
  47. G

    Integral Involving the Dirac Delta Function

  48. S

    Mathematica Derivative of the Real Part of a Complex Function (Mathematica)

    When I type in this: D [ Re[ Exp[u + 10*I] ], u ] /. u->0.5 I get this output: Of course, I could just put the Re outside and the D inside, but it would be nice to know what is wrong with the above. What's with the Re' in the output?
  49. dirtypurp

    I Is √9x a Bijection from N to R?

    Let f : N −→ R and f(x) = √ 9x The domain is all natural numbers: {0, 1, 2, 3, ...} The codomain is all real numbers. The range i believe is [0, +infinity) I believe that although the above is a function since every input of x provides a output that fits in our codomain. I also believe that...
  50. Vick

    A Associated Legendre Function of Second Kind

    The associated Legendre Function of Second kind is related to the Legendre Function of Second kind as such: $$ Q_{n}^m(z)= (-1)^m (1-z^2)^{m/2} \frac{d^m}{dz^m}(Q_{n}(z)) $$ The recurrence relations for the former are the same as those of the first kind, for which one of the relations is: $$...
Back
Top