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

    Finding a domain for a function

    I am having some trouble find the domain with this function: ##f(x)=\frac{1}{\sqrt{x^2-4x\cos(\theta)+4}}## and ##\theta\in[0,\pi]##.I know that the denominator needs to be greater than 0. So ##\sqrt{x^2-4x\cos(\theta)+4}>0##. I squared both side of the inequality, ##x^2-4x\cos(\theta)+4>0##...
  2. C

    MHB Finding the domain of this function.

    Dear Everyone, I am having some trouble find the domain with this function: $f(x)=\frac{1}{\sqrt{x^2-2x\cos(\theta)+4}}$ and $\theta\in[0,\pi]$. My attempt: I know that the denominator needs to be greater than 0. So $\sqrt{x^2-2x\cos(\theta)+2}>0$. I squared both side of the inequality...
  3. Lord Doppler

    Engineering Transfer function of AC Transformer

    I'm solving this exercise, first I did a force diagram for the transformer nucleus and I got this: ∑Fx = ma P(t) - Fk - Fb = ma P(t) = mx''+ bx' + kx So I got that dynamic equation, my question is, after transform that dynamic equation to Laplace Domain how can I relate it with the Output...
  4. S

    B Value of t for Probability Generating Function

    My questions: 1) What about if t = 2? Is there a certain meaning to ##G_X (2)## ? 2) PGF for uniform distribution is ##G_X (t)=\frac{t(1-t^n)}{n(1-t)}## and for t = 1 ##G_X (1)## is undefined so ##G_X (1) =1## is not true for all cases? Thanks
  5. N

    MHB Does the function have a zero in the given interval?

    Verify the given function has a zero in the indicated interval. Then use the Intermediate Value Theorem to approximate the zero correct to three decimal places by repeatedly subdividing the interval containing the zero into 10 subintervals. f (x) = x3 − 4x + 2; interval: (1, 2) I don't...
  6. mcastillo356

    B Derivative of the product of a function by a constant (possible typo)

    Hi, PF, I think I've found a typo in my textbook. It says: "In the case of a multiplication by a constant, we've got $$(Cf)'(x)=\displaystyle\lim_{h \to{0}}{\dfrac{Cf(x+h)-Cf(x)}{h}}=\displaystyle\lim_{h \to{0}}{\dfrac{f(x+h)-f(x)}{h}}=Cf'(x)$$" My opinion: it should be...
  7. H

    MHB Solving Function Expressions h(x): (x+a)2 +b & h-1 Range

    The function h is defined as h : x x 2 – 8x – 9 where x ≥4 (a) h(x) can be written in the form (x+a) 2 +b, find the value of ‘a’ and ‘b’ (b) Express the inverse function h –1 in the form h –1 : x ... (c) Find the range of function ‘h’.
  8. S

    Engineering Finding the natural frequency of transfer function (2s) / (3s^2+5s+2)

    In the context of control systems, if I have a vibratory second-order system, (ω_n)^2 / [s^2 + 2ζ(ω_n) + (ω_n)^2], I know how to get the natural frequency ω_n. So, if I have something like 2 / (3s^2+5s+2), I know how to get the natural frequency ω_n. However, if I instead have something like...
  9. greg_rack

    Solving an immediate indefinite integral of a composite function

    That's my attempt: $$\int (\frac{1}{cos^2x\cdot tan^3x})dx = \int (\frac{1}{cos^2x}\cdot tan^{-3}x) dx$$ Now, being ##\frac{1}{cos^2x}## the derivative of ##tanx##, the integral gets: $$-\frac{1}{2tan^2x}+c$$ But there is something wrong... what?
  10. icesalmon

    Engineering Determine the transfer function of the block diagram

    Step 1: I first started by reducing the inside of the block diagram of picture "bd" (the portion with G1 and the negative feedback G2) I obtained G1/[1 + G1G2] I'll call this term "F" Step 2: Then I'm left with two terms feeding into a summing point: F - G3 I'll call this term "K" I can...
  11. hyksos

    A Wave Function Collapse and Thermodynamic Irreversible Processes

    Very early in the development of thermodynamics, it was realized that the 2nd Law of Thermodynamics is not a law fundamental to the fabric of our cosmos, but only becomes true in the limit of the number of particles. It was none other than Boltzmann himself who realized and articulated this...
  12. P

    Proving the existence of a real exponential function

    Ok, first I tried to show that ##A = \left \{a^{r}|r\in\mathbb{Q},r<x \right \}## does not have a maximum value. Assume ##\left\{ a^{r}\right\}## has a maximum, ##a^{r_m}##. By this hypothesis, ##r_{m}<x## and ##r_{m}>r\forall r\neq r_{m}\in\mathbb{Q}##. Consider now that ## q\in\mathbb{Q}|q>0##...
  13. chwala

    Find the derivative of given function and hence find its integral

    ##y=x^2ln x-x## ##\frac {dy}{dx}=2x ln x+x-1## ##\int [2xln x+x-1]\,dx##=##x^2ln x-x##, since ##\int -1 dx= -x## it follows that, ##\int [2x ln x +x]\,dx##=##x^2 ln x## ##\int 2x ln x \,dx = x^2ln x##+##\int x\,dx## ##\int_1^2 xln x\,dx =\frac {x^2ln x}{2}##+##\frac{x^2}{4}##=##2ln2+1-0.25##
  14. J

    Passing function type as default template parameter

    The book is asking me to write my own unique_ptr template (after just covering a bit about templates). I called my template single_ptr, and I gave it two template parameters, T and D. T is supposed to be the type that the raw pointer points to. D is supposed to represent a function type so that...
  15. greg_rack

    Why Does My Integration by Parts Result Differ?

    Hi guys, I've attempted to integrate this function by parts, which seemed to be the most appropriate method... but apparently, I'm getting something wrong since the result doesn't match the right one. Everything looks good to me, but there must be something silly missing :) My attempt:
  16. J

    I Ricci scalar for FRW metric with lapse function

    I need the Ricci scalar for the FRW metric with a general lapse function ##N##: $$ds^2=-N^2(t) dt^2+a^2(t)\Big[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ d\phi^2)\Big]$$ Could someone put this into Mathematica as I don't have it?
  17. S

    Integration of this trigonometry function

    Is it possible to do the integration? That is the full question I don't know where to start, try to use ##u=\cos x## and also ##\cos^2 (x) = \frac{1}{2} + \frac{1}{2} \cos (2x)## but failed. Thanks
  18. Z

    Derivative of Cost Function with Respect to Output Layer Weight

    This is an issue I've been stuck on for about two weeks. No matter how many times I take this derivative, I keep getting the same answer. However, this answer is inevitably wrong. Please help me to understand why it incorrect. To start, I will define an input matrix ##X##, where ##n## is the...
  19. docnet

    Find the wave function of a particle in a spherical cavity

    (a) Let the center of the concentric spheres be the origin at ##r=0##, where r is the radius defined in spherical coordinates. The potential is given by the piece-wise function $$V(r)=\infty, r<a$$ $$V(r)=0, a<r<R$$ $$V(r)=\infty, r<a$$ (b) we solve the Schrodinger equation and obtain...
  20. Eclair_de_XII

    Proving continuity of inverse cube function

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  21. C

    I Expansion of a complex function around branch point

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  22. docnet

    Find the composition of a multivariate function with itself

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  23. S

    Is ideal fluid flow around an airfoil the Lambert W function?

    NOTE: By "ideal", I mean incompressible & inviscid (the forum doesn't allow a long enough title). I was looking at this Wikipedia article, and the graph of this function struck me as looking exactly like ideal fluid flow around an object. https://en.wikipedia.org/wiki/Lambert_W_function...
  24. M

    Showing that this identity involving the Gamma function is true

    My attempt at this: From the general result $$\int \frac{d^Dl}{(2\pi)^D} \frac{1}{(l^2+m^2)^n} = \frac{im^{D-2n}}{(4\pi)^{D/2}} \frac{\Gamma(n-D/2)}{\Gamma(n)},$$ we get by setting ##D=4##, ##n=1##, ##m^2=-\sigma^2## $$-\frac{\lambda^4}{M^4}U_S \int\frac{d^4k}{(2\pi)^4} \frac{1}{k^2-\sigma^2} =...
  25. M

    A Example of Ritz method with Bessel functions for trial function

    Hi PF! Do you know of any examples of the Ritz method which use Bessel functions as trial functions? I’ve seen examples with polynomials, Legendre polynomials, Fourier modes. However, all of these are orthogonal with weight 1. Bessel functions are different in this way. Any advice on an...
  26. JD_PM

    Integration and hyperbolic function problem

    This question arose while studying Cosmology (section 38.2 in Lecture Notes in GR) but it is purely mathematical, that is why I ask it here. I do not see why the equation $$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$ Has the following...
  27. Y

    Function Template: Solve Complications with Mixing Types for Variable Swapping

    Hi I want to see whether there is a way to make this program work without much complication. I read from Ivor book that you can work with two different type of variables for example mixing int x = 1 and double y = 2.5 and use template to swap them using declaration <double> in the program as...
  28. Arman777

    Is there a Python function that finds an unknown inside an integral?

    I have a integral with unknown h. My integral looks like this where C, a, b are constants F(x) and G(x) are two functions. So the only unknows in the integral is h. How can I solve it ? I guess I need to use scipy but I don't know how to implement or use which functions. Thanks
  29. Sciencemaster

    I Does the wave function spread more quickly after it is observed?

    For the sake of this question, I am primarily concerned with the position wave function. So, from my understanding, the wave function seems to 'collapse' to a few states apon measurement. We know this because, if the same particle is measured again shortly after this, it will generally remain in...
  30. Strand9202

    Velocity of an Object given its position as a function of time

    Attached is the problem and my work through the problem. I got the problem correct, but my teacher said this could be done quicker on a calculator. Any idea how it could be done quicker.
  31. Shackleford

    A Green's Function half-space

    This is from Evans page 37. I seem to be missing a basic but perhaps subtle point. Definition. Green's function for the half-space ##\mathbb{R}^n_+,## is \begin{gather*} G(x,y) = \Phi(y-x) - \Phi(y-\tilde{x}) \qquad x,y \in \mathbb{R}^n_+, \quad x \neq y. \end{gather*} What's the proper way to...
  32. JD_PM

    I Green's function for massive photon theory

    I am studying the 'toy' Lagrangian (Quantum Field Theory In a Nutshell by A.Zee). $$\mathcal{L} = - \frac{1}{4} F_{\mu \nu}F^{\mu \nu} + \frac{m^2}{2}A_{\mu}A^{\mu}$$ Which assumes a massive photon (which is of course not what it is experimentally observed; photons are massless). The...
  33. jdawg

    Total Emissivity as a Function of Temperature (Ceramics)

    Hello, I’m trying to better my understanding of how the total emissivity changes with temperature for ceramic materials. Currently it is my understanding that non-metals typically have a high emissivity. A sanded surface will result in a higher emissivity, and that spectral emissivity varies...
  34. H

    If the wave function is normalized, what is the probability density at x?

    The wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax, ψ(x) = 0 for x < 0 and x > L. When the wave function is normalized, the probability density at coordinate x has the value? (A) 2x/L^2. (B) 2x^2 / L^2. (C) 2x^2 /L^3. (D) 3x^2 / L^3. (E) 3x^3 / L^3 Ans : D
  35. Strand9202

    Derivative of the square root of the function f(x squared)

    I started out by rewriting the function as (f(x^2))^(1/2). I then did chain rule to get 1/2(f(x^2))^(-1/2) *(f'(x^2). - I think I need to go further because it is an x^2 in the function, but not sure.
  36. B

    Python Minimization likelihood function with parameters

    Hallo at all! I'm learning statistic in python and I have a problem to show you. I have this parametric function: $$P(S|t, \gamma, \beta)=\langle s(t) \rangle \left( \frac{\gamma-\beta}{\gamma\langle s(t) \rangle -\beta}\right)^2\left( 1- \frac{\gamma-\beta}{\gamma\langle s(t) \rangle...
  37. A

    I Change of variables in the Density of States function

    I have a problem where I am given the density of states for a Fermion gas in terms of momentum: ##D(p)dp##. I need to express it in terms of the energy of the energy levels, ##D(\varepsilon)d\varepsilon##, knowing that the gas is relativistic and thus ##\varepsilon=cp##. Replacing ##p## by...
  38. abhinavabhatt

    A Second derivative of Heaviside step function

    In QFT by peskin scroeder page 30 the action of Klein Gordon Operator on propagator (∂2+m2)DR(x-y)=∂2θ(x0-y0)... how to compute this ∂2θ(x0-y0)?
  39. L

    A Integral -- Beta function, Bessel function?

    Integral \int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3} Is it possible to write integral ##\int^{\pi}_0\sin^3xdx## in form of Beta function, or even Bessel function?
  40. M

    MHB What are Vieta's Formulas in Polynomial Functions?

    I say the answer is A.
  41. J

    Expressing horizontal velocity as a function of time for a wave

    This is more of a conceptual question. To find the horizontal velocity as a function of time for the above wave function, you take its partial t derivative and insert x=4. In other words the function would be -2.4sin(1-12t). Im wondering why you take the partial t derivative and not to partial...
  42. M

    MHB Well-defined function that doesn't satisfy IVT

    Hey! :giggle: Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$, $f(x):=x^2$. Show that : (i) $f$ is well-defined. (ii) $f(1)=1$, $f(3)=9$ (iii) $f$ does not satisfy the intermediate value theorem (e.g. not on $[1,3]\cap \mathbb{Q}$) For (i) do we just say that $f$ is well-defined from $\mathbb{Q}$...
  43. JD_PM

    Showing that a given propagator is proportional to Green's function

    First off let me say I am a bit confused by this question. Searching for some references I found the following related to the KG propagator, given by (P&S, chapter 2 pages 29, 30) Then they Fourier-transformed the KG propagator Is this what is aimed with this exercise? If yes, could you...
  44. LCSphysicist

    Wave function for the Helium molecule

    I am having a trouble to understand why the helium's wave function (in which we are ignoring the electric interaction between the electrons, as well the motion and problems that arise in considering the nucleus in the wave function) can be written as the product of the wave function of both...
  45. docnet

    Verify or refute the function is a solution to a PDE

    Solution attempt: We first write ##u(x)=\frac{1}{2}||x||^2## as ##u(x)=\frac{1}{2}(x_1^2+x_2^2+...+x_n^2)## Operating on ##u(x)## with ##\Delta##, we have ##u(x)=\frac{1}{2}(2+2+...+2)## adding 2 to itself ##n## times. So ##\Delta u(x)=n## and the function satisfies the first condition...
  46. MahdiI84

    I Obtain the Density of State using the Green function

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  47. F

    Four-momenta trend as a function of proper time

    As a starting point I immediately thought about the equation: ##\frac{dp^\mu}{d\tau}=qF^{\mu\nu}v_\nu## From this I proceed component by component: ##\frac{dp^0}{d\tau}=qF^{0\nu}v_\nu=q\gamma E_yv_y## ##\frac{dp^1}{d\tau}=qF^{1\nu}v_\nu=q\gamma v_yB_z##...
  48. A

    I Taylor expansion of an unknown function

    Hello, I have a question regarding the Taylor expansion of an unknown function and I would be tanksful to have your comments on that. Suppose we want to find an analytical estimate for an unknown function. The available information for this function is; its exact value at x0 (f0) and first...
  49. Y

    C/C++ Why Does My C++ Divide Function Not Compile?

    Hi I just moved into Chapter 16 of Gaddis on Exception, Templates and STL. Seems like it jump a chapter. This is a partial sample from the book: I have to complete the program as shown below but compiler doesn't like it. #include <iostream> #include <functional> using namespace std; int...
  50. N

    Separating a wave function into radial and azimuthal parts

    I know how to work through this problem but I have a question on the initial separation of the wave function. Assuming ##\psi(\rho, \phi) = R(\rho)\Phi(\phi)## then for the azimuthal part of the wavefunction we have ##\Phi(\phi)=B\left(\frac \rho\Delta cos\phi+sin\phi\right)##, but this function...
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