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. T

    A When KE is a function of position

    Hi all In the Lagrangian, we have L = KE - PE In most cases, I have seen KE as a function of q and q-dot (using the generic symbols). However I first learned how KE = 0.5 m * v-squared. Later, I used generalized coordinates and THAT is when KE became a function of q. I get all that...
  2. M

    Mathematica Fitting solution function of NDSolve with a curve

    The following solves an IVP, giving the output as the function f3[x]: s3 = NDSolve[{(-z1[t]^(3/2) + (1 + z1[t]^2)^(3/4))/( 3 (-z1[t] + Sqrt[1 + z1[t]^2])) == z1[t] z1'[t], z1[0] == 0.0001}, z1, {t, 0, 30} f3[x_] := z1[x] /. First[s3]; My question is, how do I curve fit f3[x] to the...
  3. P

    Proof that given function is convex

    Part 1 ##\left\| \vec{y} \right\|^2 \leq \left\| \vec{y} \right\|^2## and since ##\lambda \in \left[ 0,1 \right] \Rightarrow \lambda^2 \leq \lambda## So ##\lambda^2 \left\| \vec{y} \right\|^2 \leq \lambda \left\| \vec{y} \right\|^2 ## Part 2 ##\left\| \vec{x} \right\|^2 \leq \left\| \vec{x}...
  4. S

    I Derivative of F = f(x)/f(x+dx)

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  5. R

    Finding the maximum of a function

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  6. anaisabel

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  7. Z

    How to choose the correct function to use for a Taylor expansion?

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  8. brotherbobby

    Line integral of a scalar function about a quadrant

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  9. mcastillo356

    I A function with no max or min at an endpoint

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  10. R

    Fourier series, periodic function for a string free at each end

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  11. Hamiltonian

    I Writing the wave function solutions for a particle in a 2-D box

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  12. LUFER

    I Plasmonic - Dielectric function

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  13. P

    A $\phi^4$ in $4 - \epsilon$ dimension renormalization beta function

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  14. L

    I Second derivative of chained function

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  15. patric44

    Checking if a function is an equipotential surface

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  16. Physil

    An expression for the vertical velocity as a function of time

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  17. S

    MHB Interpolating Points with Continuous Modular Functions?

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  18. MechEEE

    Transfer function with initial conditions (DE)

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  19. AncientOne99

    Engineering Electrical and Control Engineering: Transfer Function Reduction problem

  20. S

    Using the sin function for a problem with a frictionless pulley and an incline

    To find the tension in the rope connecting 6.0 kg block and 4.0 kg block we do 6.0 kg = m1, 4.0 kg = m2, 9.0 kg = M (m_2 + m_1)a - Ma = Mg - m_2 gsin\theta - m_1 gsin\theta Why do we use sin in these equations and not cos?
  21. docnet

    Prove that a function from [0,1] to [0,1] is a homeomorphism

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  22. docnet

    Prove that the concatenation function is continuous

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  23. Mikaelochi

    I Doing proofs with the variety function and the Zariski topology

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  24. M

    A Array variable of envelope function (parameter representation)

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  25. Vividly

    I Question about Inverse Derivative Hyperbola function

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  26. I

    How can I plot the function g(x) = sin(πn/L) x and its corresponding g²(x)?

    Summary:: We are currently studying basics of quantum mechanics. I'm getting the theory part but it's hard to visualise everything and understand. We are given this question to plot the function so if someone could help me in this. Plot the following function and the corresponding g²(x) g(x)...
  27. LCSphysicist

    Find the intensity as function of y (interference between two propagating waves)

    Let a spherical wave propagate from the origin, $y = ADcos(wt-2\pi r/ \lambda)/r$. Also, let a plane wave propagate parallel to the x axis, $y = Acos(wt-2\pi r/ \lambda)$. At x = D there is a flat screen perpendicular to the x axis. Find the interference at the point y on the screen as function...
  28. laserdan

    A Rate and function to fill a theoretical vacuum

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  29. Jarvis323

    I Which kind of function is this?

    I'm curious how close someone could get to guessing the functions that generated the data shown below. And also, without looking at the plot, what do you think would be the most interesting looking function of x,y,z you can think of. A) B) C)
  30. Mikaelochi

    I Proving a function f is continuous given A U B = X

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  31. F

    "Trick" for a specific potential function defined with an integral

    Hello, To first clarify what I want to know : I read the answer proposed from the solution manual and I understand it. What I want to understand is how they came up with the solution, and if there is a way to get better at this. I have to show that, given a vector field ##F## such that ## F ...
  32. docnet

    Prove that a locally constant function is constant on a connected X

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  33. R

    How to find the residue of a complex function

    Hi, I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$ Since I have 2 singularities which are double poles. I'm using this formula $$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$ then, $$\lim_{z\to\ \pm ia}...
  34. Mayhem

    Particle in a box: Finding <T> of an electron given a wave function

    If ##\hat{T} = -\frac{\hbar}{2m}\frac{\mathrm{d^2} }{\mathrm{d} x^2}##, then the expectation value of the kinetic energy should be given as: $$\begin{align*} \left \langle T \right \rangle &= \int_{0}^{L} \sqrt{\frac{2}{L}} \sin{\left(\frac{\pi x}{L}\right)}...
  35. J

    Is the magnetic field B→. a state function and exact differential?

    is the magnetic field B→. a state function and exact differential? I argued that it's a state function, what do you guys think
  36. chwala

    Find the domain of the inverse of a function

    This is a textbook problem: now for part a) no issue here, the range of the function is ##-1≤f(x)≤299## now for part b) i got ##x≥-1## but the textbook indicates the solution as ##x≥0## hmmmmm i think, that's not correct...
  37. Arman777

    I What is the function that describes this Asymptotic behaviour?

    I would like to find a function such that for $$a(x) \rightarrow 1~\text{for}~(x \gg x_c)$$ $$a(x) \rightarrow f(x)~\text{for}~(x \ll x_c)$$ What could be the ##a(x)## ? I have tried some simple functions but could not figure it out. Maybe I am just blind to see the correct result.
  38. K

    A Dissipation function is homogeneous in ##\dot{q}## second degree proof

    We have Rayleigh's dissipation function, defined as ## \mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right) ## Also we have transformation equations to generalized coordinates as ##\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}...
  39. PainterGuy

    Random variable and probability density function

    Hi, I was trying to solve the attached problem which shows its solution as well. I cannot understand how and where they are getting the equations 3.69 and 3.69A from. Are they substituting the values of θ₁ and θ₂ into Expression 1 after performing the differentiation to get equations 3.70 and...
  40. tixi

    Finding analyticity of a complex function involving ln(iz)

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  41. O

    Arc length of vector function - the integral seems impossible

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  42. K

    Sketch the function by hand -- I'm confused on how to do this

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  43. PainterGuy

    I Distribution function and random variable

    Hi, I cannot figure out how they got Table 2.1. For example, how come when x=1, F_X(x)=1/2? Could you please help me with it? Hi-resolution copy of the image: https://imagizer.imageshack.com/img923/2951/w9yTCQ.jpg
  44. R

    Cauchy Riemann complex function real and imaginary parts

    Hi, I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}## First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}## Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}## thus, ##\frac{df}{dz} =...
  45. L

    I Understanding relationship between heat equation & Green's function

    Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is: $$ u(x, t) = \int \phi(x-y, t)f(y)dy $$ where ##\phi(x, t)## is the heat kernel. The integral looks a lot similar to using Green's function to solve differential...
  46. Fred Wright

    A What is the unique special function in this integral problem?

    While studying the solution to a integral problem I found online I ran across a special function I am unfamiliar with. The integral is $$ \int_0^{\infty}\frac{t^{\frac{m+1}{n}-1}}{1+t}dt=\mathcal{B}(\frac{m+1}{n},1-\frac{m+1}{n}) $$ This certainly isn't the normal beta function. What is it...
  47. Eclair_de_XII

    B Is it invalid to redefine the sgn function in this way in a proof?

    Let ##S_n## denote ##\{1,\ldots,n\}##, where ##n\in\mathbb{N}##. Recall that the ##\textrm{sgn}## function maps a permutation of ##S_n## to an element in ##\{1,0,-1\}##. We want to rework the definition of ##\textrm{sgn}## because it is not sufficient for some proofs about determinants. For...
  48. Dom Tesilbirth

    How to find the partition function of the 1D Ising model?

    Attempt at a solution: \begin{aligned}Z=\sum ^{N}_{r=0}C\left( N,r\right) e^{-\beta \left[ -NJ+2rJ\right] }\\ \Rightarrow Z=e^{\beta NJ}\sum ^{N}_{r=0}C\left( N,r\right) e^{-2\beta rJ}\end{aligned} Let ##e^{-2\beta J}=x##. Then ##e^{-2\beta rJ}=x^{r}##. \begin{aligned}\therefore Z=e^{\beta...
  49. M

    I Fit a non-linear function to this time series

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  50. Poetria

    Matrix Function - Check Understanding

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