Function Definition and 1000 Threads

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. A

    I Function value at different points

    If for a field say ##f=xyz## ,it's value is defined at 10 points in the 3-D Cartesian co-ordinate system...now using these 10 values of f and the corresponding coordinates is it possible to find the value of f at any ##(x,y,z)## of choice??
  2. V

    A What type of function satisfy a type of growth condition?

    Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established: \begin{equation} ||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right), \end{equation} with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
  3. E

    Compute the energy for 2D wave function with discontinuous derivatives

    I have calculated the normalization constant, but I'm struggling with the discontinuities in the derivatives of the wave function. Due to the symmetry, it should suffice to consider the first two cases. The results should be (according to WolframAlpha): \left( \frac{\partial^{2}}{\partial...
  4. D

    I Changing the argument of a function

    Hi. If I have a function f(x) = √(x+1) and I define u=x+1 is it correct to state f(u) = √u ?
  5. QuasarBoy543298

    I Particle in free space - what happens to the wave function after measurement?

    If I'm trying to solve the problem of a particle in free space (H = P2/2m ). the eigenfunctions of the Hamiltonian cannot be normalized. now assume I have a legitimate wave function expressed in terms of the eigenfunction of H and I want to measure its energy. what will happen to the...
  6. R

    I Arc diameter as a function of arc length and chord length

    I'm trying to determine if a certain bicycle tire size will fit my bike, and that determination is based on the inflated diameter (or width) of the tire. As such, I'm trying to come up with a formula that will give me the diameter of a bicycle tire as a function of the tire's carcass width and...
  7. R

    Troubleshooting Jump Discontinuities: Causes & Solutions

    Here is a picture of these plots from a paper: When I try to reproduce the 3rd graph above (yellow line below), I get sharp discontinuities: Those jump discontinuities should not occur, and the function should never rise to the high value of the two other plots. So, what could be the cause...
  8. mishima

    Sn(u), Jacobi elliptic function, for simple pendulum of any amplitude

    I understand how to reach $$\int_0^\phi \frac{d\theta}{\sqrt{1-k^{2}sin^{2}\theta}}=\sqrt \frac g l t$$ from physics but from there I don't get how to turn that into this new (for me) sn(u) form.
  9. Rahulx084

    Questions about the Point Function (Thermodynamics)

    We know from first law of thermodynamics for a closed system that ##dE##=##\delta Q## -##\delta W## , my question is that for a closed adiabatic system net heat transfer =0 this mean net change in energy = work done , does that mean for an adiabatic system work done is a point function as...
  10. humancentered666

    What Exactly Does Equation (2) Mean? (Equations of Motion from PE function)

    What exactly is this equation telling me? How can I use it to work out the Equations of Motion given an equation of potential energy? Won't I have to solve a PDE? I'm extremely sorry if this question comes off ignorant.
  11. shahbaznihal

    I Phase space density function and Probability density function

    I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of Gravitational Focusing of WIMPs at the location of the Earth due to the gravitational well of the Sun...
  12. A

    A Solving analytic gradient for multilayer perceptron loss function

  13. NP04

    Finding the limits of a piecewise function

    Problem Statement: Determine whether f is continuous at c. (see image for piecewise function f) EDIT: Sorry if it is a little blurry that is x^3 in the numerator of the rational function and x^2 in the denominator Relevant Equations: Basic understanding of limits My work: Since the...
  14. shahab44

    A Replacing a non-harmonic function with a harmonic function

    I am solving a problem of the boundary condition of Dirichlet type, in order to solve the problem, the functions within the differential equations suppose to be harmonic. I have a function f(x,y,z) (the function attached) which is not harmonic. I must find an equivalent function g(x,y,z) which...
  15. S

    Find the Taylor series of a function

    Because the Taylor series centered at 0, it is same as Maclaurin series. My attempts: 1st attempt \begin{align} \frac{1}{1-x} = \sum_{n=0}^\infty x^n\\ \\ \frac{1}{x} = \frac{1}{1-(1-x)} = \sum_{n=0}^\infty (1-x)^n\\ \\ \frac{1}{x^2} = \sum_{n=0}^\infty (1-x^2)^n\\ \\ \frac{1}{(2-x)^2} =...
  16. dRic2

    I Green's function for the wave equation

    Hi, I'm reading "Wave Physics" by S. Nettel and in chapter 3 he introduces the Green's function for the 1-dimensional wave equation. Using the separation of variables method he restricts his attention to the spatial component only. Let ##u(x)## be the spatial solution to the wave equation and...
  17. C

    Gradient of a function at many points, referencing a struct

    I want to compute the gradient of some smooth function at many points by taking the value of the function at point x(i) subtracted from the value of the function at point x(i+1) and then divide the result by ( x(i+1)-x(i) ). My function has a struct as an argument and within that struct I have...
  18. D

    Drawing a derivative of a function

    Red line being the function and blue an approximation of the derivative. Does it look right?
  19. Ramil

    Quantum Monte-Carlo calculation of Green's function

    Introducing the spacetime spherical symmetric lattice, I use the following notifications in my program. i - index enumerating the nodes along t-coordinate, j - along the r-coordinate, k - along the theta-coordinate, l - along the phi-coordinate. N_t - the number of nodes along t-coordinate. N_r...
  20. A

    Three dimensional ##\delta## function

    ##r,\theta,\phi## are the usual spherical polar coordinate system. ##\int_v\nabla•(\frac{\hat r}{r})dv## over a spherical volume of radius ##R## reduces to ##\int_s(\frac{\hat r}{r})•\vec ds=4\pi R## Now ##r## runs from 0 to ##R,\theta## from 0 to ##\pi## and ##\phi## from 0 to ##2\pi##. In...
  21. Robin04

    Calculating the residue of a complex function

    The singularities occur at ##z = \pm i\lambda##. As ##\frac{d}{dz}(z^2+\lambda^2)^2|_{z=\pm i\lambda}=0##, these singularities aren't first order and the residues cannot be calculated with differentiating the denominator and evaluating it at the singularities. What is the general method to...
  22. Terrycho

    The Potential Energy Function in Three-Dimensional Motion

    I set the location of the particle (x,y,z); therefore, → the force F_1 is (z^2/root(x^2+y^2) * x/root(x^2+y^2) , z^2/root(x^2+y^2) * y/root(x^2+y^2), 0), since cosΘ is x/root(x^2+y^2). →...
  23. amjad-sh

    Using Simpson's method to integrate a complex function

  24. R

    Calculating the half maximum point of a function

    This is the form of the function above: I started by equating (1) to 1/2: $$T(\varphi)=\frac{r^{2}+\tau^{2}-2\tau\cos\varphi}{1+\tau^{2}r^{2}-2\tau r\cos\varphi} = \frac{1}{2},$$ which can be rearranged to: $$2r^{2}+2\tau^{2}-1-\tau^{2}r^{2}=2\tau\left[2-r\right]\cos\varphi$$ using...
  25. S

    MHB Improper integral of an even function

    Hi colleagues This is a very very simple question I can show when $f$ is integrable and is even i.e. $f(-x)=f(x)$ then $\int_{-a}^{a} \,f(x)\,dx=2\int_{0}^{a} \,f(x)\,dx$ what about improper integrals of even functions, like the function ${x}^{2}\ln\left| x...
  26. D

    Need help with Matlab Function of Differential Equations

    WHAT HAPPENS IS That I need to model the example of A Protein G example, using a function f in Matlab, but when I execute the script, the graphics I get do not correspond to those of the example. The problem is that I can not understand what the model seeks to represent, besides that I do not...
  27. M

    A What are the equality conditions for proving strict convexity?

    Hi PF! Do you know what a strictly convex function is? I understand this notion in the concept of norms, where in the plane I've sketched the ##L_1,L_2,L_\infty## norms, where clearly ##L_1,L_\infty## are not strictly convex and ##L_2## is. Intuitively it would make sense that any...
  28. tj77

    Y directed force as a function of time

    The question before asked to find the net force as a function of time which I got: F = 4.83×10¹⁰ (1.65×10⁻⁸ t − 7.41×10⁻⁶) N I just have no idea how to do it with the y directed force since I only have a horizontal acceleration equation. Thanks so much to anyone that helps, I appreciate it!
  29. Samwell

    How long will it take for an object to stop with a defined force over time?

    Hello, I have the force defined as a function of time, where F=A-Bt and A=100N, B=100Ns-1. I have to determine, how long it will take for object to stop, if t0=0s and v0=0,2ms-1 and mass of the object is m=10kg. Can somebody please help me with this, because I'm having hard time with this task.
  30. D

    Gradient & Smooth Surfaces: Implicit Function Theorem

    Section ##3.8## talks about the gradient and smooth surfaces, defining when the directional derivative ##(\partial f/\partial\mathbf{u})(\mathbf{p})## takes maximum value and that when it equals ##0##, then ##\mathbf{u}## is a unit vector orthogonal to ##(grad\ f)(\mathbf{p})##.It also says that...
  31. M

    Find a function less than a fraction of itself

    Well doesn't ##u(x) = 0.4 x## work? Seems too easy, but the phrasing at the end "for all ##x\in I##" makes me think since ##0.4x = x## only at ##x=0##, and not all of ##I##, that this is okay. But am I wrong?
  32. C

    Net force as a function of time?

    All I've done so far is think about F_net. Since F=ma, and a is a vector, I was thinking that I should find the x and y components of a and then try to calculate F_net that way, but I'm confused as to where I should use x(t) and y(t). Or instead, thinking about it as the change in momentum over...
  33. cookiemnstr510510

    What happened to simple function?

    Hello All, I have a question regarding the simple function in MATLAB. My textbook talks about it and it looks very useful, it will show you a bunch of steps of how to simplify an expression or equation. I am using MATLAB R2018b and it looks like the function is gone. I am wondering if something...
  34. R

    MHB Finding Discontinuities: Multiply by Conjugate?

    I recently had to find what f(7) equals if f(x) = \frac{x^2-11x+28}{x-7}. I first tried \frac{x^2-11x+28}{x-7} \cdot \frac{x+7}{x+7}, and it seemed like a perfect fit since I eventually got to \frac{x^2(x-4)-49(x+4)}{x^2-49}=(x-4)(x+4), but that gave me f(7)=33, instead of the right answer...
  35. J

    A Studying Green's function in many body physics

    Hi,everyone. Recently, I am studying green's function in many body physics and suffer from trouble.Following are my problems. (1) What is the origin of the definition of green's function in many body physics? (2) What is the physical meaning of self energy ? It seems like it is the correction...
  36. J

    A Why "Green's function" is used more than "correlations" in QFT?

    Very often, the term "Green's function" is used more than "correlations" in QFT. For example, the notation: $$<\Omega|T\{...\}|\Omega> =: <...>$$ appears in Schwartz's QFT book. And it seems very natural, basically because the path integral definition of those terms "looks like" the...
  37. Q

    A One Loop Correction to a 4 pt. function in 3 dimensions

    If I have a Lagrangian of the form \mathcal{L}=-\frac{1}{2} (\partial \phi)^2 - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{3!} \phi^6, in 3 dimensions, what is the one-loop correction to the 4-point function? Am I correct in thinking that the following Feynman diagram is the representation of the...
  38. W

    I Newton-Raphson in Least Squares: How is it used? Cost Function?

    I just went over analysis of a data set that was analzed using Linear Regression (OLS, I believe) and I saw Newton's method was used. Just curious, how is it used? I assume to minimize the cost function, but this function was not made explicit. Anyone know? Thanks.
  39. S

    Positive derivative implies growing function using Bolzano-Weierstrass

    I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement: $$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the...
  40. karush

    MHB 141.30 how many points of inflection will the graph of the function have

    If the derivative of a function f is given by $$f'(x)=\frac{1}{5}(x^2-4)^5-x^2$$ how many points of inflection will the graph of the function have?solution find $f"(x)$ $$f''(x)=2x((x^2-4)^4-1)$$ at $f''(x)=0$ we have factored $$2 x (x^2 - 5) (x^2 - 3) (x^4 - 8 x^2 + 17) = 0$$ then...
  41. J

    A Which operator corresponds to the Green function in QFT?

    The Feynman propagator: $$D_{F}(x,y) = <0|T\{\phi_{0}(x) \phi_{0}(y)\}|0> $$ is the Green's function of the operator (except maybe for a constant): $$ (\Box + m^2)$$ In other words: $$ (\Box + m^2) D_{F}(x,y) = - i \hbar \delta^{4}(x-y)$$ My question is: Which is the operator that...
  42. paulmdrdo

    Maximum Amplitude of a Function

    I was able to find the maximum value for this function by differentiating and equating it to zero and find the time t and substitute it back to the original expression to get the max amplitude. tm = -0.001012 s v(tm) = 56.6 Another method that was presented in my book was can you explain how...
  43. Akash47

    Finding f(6) from a composite function

    It is obvious that the function f is not injective. From the given equation, we get f(f(2))=6.And since,there is an inequality given in the problem, I think we can use that to find f(6).But I have got stuck here and can't move.Do I have to find what is f(x) first?Then how?
  44. karush

    MHB 205.8.9 Find the derivative of the function

    205.8.9 Find the derivative of the function $y=\cos(\tan(5t-4))\\$ chain rule $u=\tan(5t-4)$ $\frac{d}{du}\cos{(u)} \frac{d}{dt}\tan{\left(5t-4\right)}\\$ then $-\sin{\left (u \right )}\cdot 5 \sec^{2}{\left (5 t - 4 \right )}\\$ replacing u with $\tan(5t-4)$ $-\sin{(\tan(5t-4))}\cdot 5...
  45. J

    Finding range of a function using inequalities

    My attempt : Given ##f(x)## and ##g(x)## for ## -1.6 < x < 1.6## we get ##0\leq f(x)<1.6## Thus, for ##f(g(x))## we get ## -3 \leq g(f(x)) < -1.4## Thus the required set should be the interval ##[-3, -1.4)##? My Questions : 1. What have I missed since my answer does not match the given...
  46. Silvio Macedo

    I Wave function of particle / quantum field in space, also in time?

    Quantum fields have wave functions that determine a particle position in space. It solves non-locality, double-slit paradox, tunnel effect, etc. What if the wave function is also in time? Won't it solve the breaking of causality at quantum level? (Delayed Choice/Quantum Eraser/Time) Not much...
  47. M

    What are the units of the argument "x" for this cos(x) function integral?

    Show that the value of ##\int_0^1\sqrt(1-cosx)dx## is less than or equal to ##\sqrt2## ##1\ge cos x\ge-1## The problem is a worked one but I am just confused by a simple thing. We integrate the function f ##\int_0^1\sqrt(1-cosx)dx in the interval [0,1] but I don't understand that what stands...
  48. J

    Finding the range of a function when checking if it is bijective

    To check if it is injective : ##h'(x) = 3(x^2-1)## ##\implies h'(x) \geq 0## for ##x \in (-\infty, -1]## Thus, ##f(x)## is increasing over the given domain and thus is one-one. To check if it is surjective : Range of ##f(x) = (0, e^4]## but co-domain is ##(0, e^5]## thus the function is into...
  49. C

    Use the output of a function in another function

    Say I have a function `func` that does a certain task, returning some expression `exp`. Can I use this expression in another function without having to call `func` again, which I suppose will redo all the steps needed to derive `exp` in the first place? E.g double func() {...
  50. J

    B Correlation between shifting graph of a function and shifting the axes

    1.To shift the graph of a function : Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative. Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is...
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