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. C

    MHB Implicit function theorem for f(x,y) = x^2+y^2-1

    $f: \mathbb{R^2} \rightarrow \mathbb{R}$, $f(x,y) = x^2+y^2-1$ $X:= f^{-1} (\{0\})=\{(x,y) \in \mathbb{R^2} | f(x,y)=0\}$ 1. Show that $f$ is continuous differentiable. 2. For which $(x,y) \in \mathbb{R^2}$ is the implicit function theorem usable to express $y$ under the condition $f(x,y)=0$...
  2. qbar

    A Implicitly differentiating the vanishing real part of the hyperbolic tangent of one plus the square of the Hardy Z function

    Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...
  3. wolfy

    B When the wave function collapses, how long is it collasped?

    When wave function collapses how long is it collasped... Shooting electrons at a double slit and observing the electrons before they reach the 2 slits collasped the wave function...so is its behavior particle like forever? Quantum mechanics is simple however wrapping ones head around it is...
  4. A

    A Identification of a probability density function

    I have the following probability density function: $$f(x) =...
  5. anemone

    MHB Finding $f(84)$ with the Defined Function $f$

    The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3, & \text{if} \,\,n\geq 1000 \\ f(f(n+5)), & \text{if}\,\, n< 1000 \end{cases} \] Find $f(84)$.
  6. Kaguro

    Finding the potential function from the wavefunction

    I would differentiate this twice and plug it into the S.E, but for that I'll need E. Which I don't have. Please provide me some direction.
  7. M

    B Critical point of a piecewise function

    If the function is not differentiable at point. Can we consider this point is critical point to the function? f(x) = (x-3)^2 when x>0 = (x+3)^2 when x<0 he asked for critical points in the closed interval -2, 2
  8. Daniel Lima

    Python How to plot a function with multiple parameters on the same set of axes

    I attached a file with some explanations of the variables in the code and the plot that I should get. I don't know what is wrong. Any help will appreciated. from scipy.integrate import quad import numpy as np from scipy.special import gamma as gamma_function from scipy.constants import e...
  9. O

    MHB Evaluate some kind of gamma function

    My question and solution that I've tried out are in attachment. Is it true my steps?
  10. F

    Which sigma algebra is this function a measure of?

    Suppose ##\nu## is a measure on some ##\sigma##-algebra ##\mathcal{A}##. Then we must have for all ##A \in \mathcal{A}## either ##A## or ##A^c## is finite, but not both. Because otherwise ##\nu(A)## is undefined or not well defined. I've verified that ##\lbrace \emptyset, X \rbrace## and...
  11. S

    Laplace transform of an ODE with a non-smooth forcing function

    Suppose I'm solving $$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?
  12. agnimusayoti

    Indefinite integral of cross product of 2 function

    I've tried with this work in attachment. i&m not sure of my answer is correct.
  13. Terrycho

    I The wave function in the finite square well

    Hello! I have been recently studying Quantum mechanics alone and I've just got this question. If the potential function V(x) is an even function, then the time-independent wave function can always be taken to be either even or odd. However, I found one case that this theorem is not applied...
  14. peelgie

    Prove that this Function is a Homomorphism

    Summary:: Abstract algebra I have a problem with this task. Please help. [Moderator's note: Moved from a technical forum and thus no template.]
  15. P

    Plotting a Bessel Function for Diffraction (Fraunhofer)

    From my understanding of diffraction pattern is supposed to result in something like this However when I plot it I get the central peak without the ripples (even when broadening the view). My result My code is as follows %1) Define the grid. Define vectors so that they include 0...
  16. Y

    MHB Objective function of a linear program with multiple variables.

    Hello, Please I need help to find the objective function of a linear program (attachement : example). I tried to figure it out from the formula provided in (attachement : formula) but I couldn't understand it, it's written (MIN(lambda)wj) I think it's the key to resolve my question ! ( Full file...
  17. Y

    Objective function of a linear program with multiple variables

    Hello, Please I need help to find the objective function of a linear program (attachement : example). I tried to figure it out from the formula provided in (attachement : formula) but I couldn't understand it, it's written (MIN(lambda)wj) I think it's the key to my question ! ( Full file is...
  18. Eclair_de_XII

    Finding a local max/min of a function in Python

    Okay, so my algorithm looks something like this: ==== 1. Locate mid-point of the interval . This is our estimate of the local max. 2. Evaluate . 3. Divide the main interval into two subintervals: a left and right of equal length. 4. Check to see if there is a point in the interval such that ...
  19. G

    MHB How Can I Solve a Problem Using Euler's Totient Function for Odd Prime Numbers?

    Hello everyone, can anybody help me with this problem? The solution is for all odd prime numbers, but I have no idea how to solve it. Any help will be greatly appreciated.
  20. A

    A How to transform a probability density function?

    I have the following probability density function (in Maple notation): f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi] Now I want to transform x so that 0 -> (3/2) * Pi and 3 * Pi -> (15/2) * Pi and the new function is still a probability density function. How should I...
  21. samuelfarley

    Calculus problem: Questions about the function f (x) = - x / (2x^2 + 1)

    shown in attachment
  22. leticia beira

    Finding the derivative of this trig function

    Para f (θ) = √3.cos² (θ) + sen (2θ), uma inclinação da reta tangente, uma função em θ = π / 6, é?
  23. jisbon

    Engineering Finding the transfer function for this circuit

    Transformed circuit: Using KVL, Now, I am unsure about the current to use KVL in this case. As far as equation goes: Vi(s) =(I1*R)+(I3*R)+Vc(s), where Vc(s) = V0(s)/u as shown in the circuit. How am I supposed to find the current I1 and I3 for the two resistors in this case? Thanks
  24. S

    Question about asymptotes of rational function

    I tried graphing the function in the calculator, and the graph seems to have a horizontal asymptote at y=0, not at y=1. Why is this so? Thanks for helping out.
  25. agnimusayoti

    How to model a function of a box's volume using Lagrange multiplier methods

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  26. A

    A How to change the support of a probability density function?

    Given the support [a, b] of a probability density function. How can I change the formula for the probability density function with a support [u, v]? Example: Given the beta distribution with support [a=0,b=1]: $$\frac{x^{p-1} (1-x)^{q-1}}{Beta(p,q)}$$ Then the beta distribution with support...
  27. anemone

    MHB Evaluate the constant in polynomial function

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  28. PGaccount

    I Partition function of quantum mechanics

    In quantum mechanics, we have the partition function Z[j] = e-W[j] = ∫ eiS+ jiOi. The propagator between two points 1 and 2 can be calculated as ## \frac{\delta}{\delta j_1}\frac{\delta}{\delta j_2} Z = \langle O_1 O_2 \rangle## The S in the path integral has been replaced by S → S + jiOi...
  29. Dwightun

    Maple How to get my function from these dsolve results

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  30. BlueTempus

    ODE -> Transfer Function Assistance

    Homework Statement:: ODE -> Transfer Function Assistance Relevant Equations:: Newtonian physics, buoyancy, drag [Mentor Note -- thread moved to DE from the schoolwork forums, since it is for work and not schoolwork] Hello all, I'm new here but I'm looking for a bit of guidance with a...
  31. anemone

    MHB Can You Solve This Tricky Trigonometric Floor Function Equation?

    Solve $\{ \sin \lfloor x \rfloor \}+\{ \cos \lfloor x \rfloor \}=\{ \tan \lfloor x \rfloor \}$ for real solution(s).
  32. A

    MHB Question about Successor Function

    One of the Peano Axioms specifies Sa = Sb --> a = b where S is the successor function. How does one establish from the axioms that S is, in fact, a function, that is the converse a = b --> Sa = Sb? Probably a very simple matter, but I would appreciate any help in clarifying. Many thanks...
  33. A

    I Need help developing a movement function for motor motion

    Hi, I have a motor that i would like to rotate to a certain angle, in a controlled manner. During the movement, i want to update the final position I want to reach. The new updated function has to start with the same speed the initial function ended with I wan to find a function that does this...
  34. tworitdash

    A Integrating a function of which poles appear on the branch cut

    I have a complicated function to integrate from -\infty to \infty . I = \int_{-\infty}^{\infty}\frac{(2k^2 - \Omega^2)(I_0^2(\Omega) + I_2(\Omega)^2) - \Omega^2 I_0(\Omega) I_2(\Omega)}{\sqrt{k^2 - \Omega^2}} \Omega d\Omega Where I0I0 and I2I2 are functions containing Hankel functions as...
  35. S

    Engineering How does feedback affect the transfer function of an integrating block?

    The correct solution is different than my answer, I am not sure where I am going wrong?
  36. P

    Confirming Green's function for homogeneous Helmholtz equation (3D)

    Plugging in the supposed ##G## into the delta function equation ##\nabla^2 G = -\frac{1}{4 \pi} \frac{1}{r^2} \frac{\partial}{\partial r} \left(\frac{r^2 \left(ikr e^{ikr} - e^{ikr} \right)}{r^2} \right)## ##= -\frac{1}{4 \pi} \frac{1}{r^2} \left[ike^{ikr} - rk^2 e^{ikr} - ike^{ikr} \right]##...
  37. Adesh

    I Checking the integrability of a function using upper and lowers sums

    Hello and Good Afternoon! Today I need the help of respectable member of this forum on the topic of integrability. According to Mr. Michael Spivak: A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if $$ sup \{L (f,P) : \text{P belongs to the set of...
  38. chwala

    Find the sum of a function given a series

    since the first term is ##g(0)= \frac {1}{3}## & last term is ##g(1)=\frac {4}{6}## it follows that the ##\sum_{0}^1 g(x)##= ##\frac {1}{3}##+##\frac {4}{6}=1## is this correct?
  39. tworitdash

    A Integral of a sinc squared function over a square root function

    I want to find the analytical solution to the integral given below. \int_{-\infty}^{\infty} \frac{ sinc^2(\frac{k_yb}{2})}{\sqrt{k^2 - k_x^2 - k_y^2}}dk_y In other words, \int_{-\infty}^{\infty} \frac{ \sin^2(\frac{k_yb}{2})}{(\frac{k_yb}{2})^2\sqrt{k^2 - k_x^2 - k_y^2}}dk_y Can this be...
  40. WMDhamnekar

    MHB Increasing and decreasing interval of this function |e^x+e^{-x}|

    Hello, I want to know what is the incresing and decreasing interval of this even function $|e^x+e^{-x}|?$ If any member knows the correct answer, may reply to this question.
  41. L

    MHB Integral limits when using distribution function technique

    I am not sure about finding the limit of the integral when it comes to finding the CDF using the distribution function technique. I know that support of y is 0 ≤y<4, and it is not a one-to-one transformation. Now, I am confused with part b), finding the limits when calculating the cdf of Y...
  42. A

    MHB Prove Monotony of Function: $f$ Strictly Decreasing

    Let $f$ be differentiable from $(-\inf,0)$ to $(0,\inf)$ and let $f'(x)<0$ for all real numbers except 0 and $f'(0)=0$. Prove that f is strictly decreasing.
  43. J

    Reducing Bessel Function Integral

    I tried integration by parts with both ##u = x^2, dv = J_0 dx## and ##u = J_0, du = -J_1 dx, dv = x^2 dx.## But neither gets me in a very good place at all. With the first, I begin to get integrals within integrals, and with the second my powers of ##x## in the integral would keep growing...
  44. ttpp1124

    Determine the equation of the tangent line to the function given

  45. 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)...
  46. 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...
  47. 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)...
  48. 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...
  49. 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 =...
  50. 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...
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