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. karush

    MHB 1.8.4 AP Calculus Exam Integral of piece-wise function

    image due to macros in Overleaf ok I think (a) could just be done by observation by just adding up obvious areas but (b) and (c) are a litte ? sorry had to post this before the lab closes
  2. Addez123

    Prove that any function z = f(x + y) solves the equation z'x - z'y = 0

    How about I prove that to be false instead? $$z = x$$ $$z'_x(x + y) = 1$$ $$z'_y(x + y) = 0$$ $$z'_x - z'_y = 1$$
  3. K

    I Derivative of a function is equal to zero

    Suppose: - that I have a function ##g(t)## such that ##g(t) = \frac{dy}{dt} ##; - that ##y = y(x)## and ##x = x(t)##; - that I take the derivative of ##g## with respect to ##y##. One one hand this is ##\frac{dg}{dy} = \frac{dg}{dx}\frac{dx}{dy} = \frac{d^2 y}{dxdt}\frac{dx}{dy}##. On the other...
  4. M

    Mathematica Finding local maxima from interpolated function

    Hi PF! I have data that I need to interpolate (don't want to go into details, but I HAVE to interpolate it). I'm trying to find the local maximas on a given domain. I've looked everywhere and still haven't been able to do it? Seems most people work with NDSolve, but I don't use that function...
  5. DEvens

    I Wave function of a laser beam?

    Summary:: Wave function of a laser beam before it hits the diffraction grating So I'm reading "Foundations of Quantum Mechanics" by Travis Norsen. And I've just read Section 2.4 on diffraction and interference. And he derives a lovely formula for the wave function of a particle after it leaves...
  6. A

    Engineering Transfer function to block diagram

    I am given the following two equations: and where E_1 is an output with corresponding input e and theta_o is an output with corresponding input E_3. The solutions that I was given are as follows: Unfortunately, I do not understand at all how to work out the block diagrams from the equations...
  7. Diracobama2181

    Help Understanding Response Function $$H(\omega)$$

    $$<H(\omega)>=\sum_{j} χ_{HAj}h_j(\omega)$$ Where $$χ_{HA}=\frac{1}{2\hbar} Tr{{\rho}[{H(t)},{A(0)}]}$$. But $$[H(t),A(0)]=[H_o,A(0)]-[A(t)h,A(0)]=-h_0 cos(\omega t)[A(t),A(0)]$$. So $$χ_{HA}=-\frac{1}{2\hbar}Tr(\rho h_0 cos(\omega t)[A(t),A(0)])=-h_0cos(\omega t)χ_{AA}$$. Then...
  8. Alexan

    Please can I get some help finding the function of motion

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  9. W

    Gamma Function Q from Mary Boas 2nd ed (ch11)

    So using $$L=\frac{mv^2}{2} - \frac{1}{2} m lnx$$ and throwing it into the Euler-L equation I agree with kcrick & OlderDan that we can manipulate this to either $$\frac{d}{dt} m\dot{x} = -\frac{m}{2x}$$ or $$2vdv = -\frac{dx}{x}$$ but I'm not having any epiphanies on how to turn the above into...
  10. Arman777

    Subscript problem for greek letters in python print function

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  11. karush

    MHB Finding the Derivative of the Inverse Function of a Cubic Polynomial

    Let $f(x)=(2x+1)^3$ and let g be the inverse of $f$. Given that $f(0)=1$, what is the value of $g'(1)?$ ok not real sure what the answer is but I did this (could be easier I am sure} rewrite as $y=(2x+1)^3$ exchange x and rename y to g $x=(2g+1)^3$ Cube root each side...
  12. U

    Probability generating function

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  13. Y

    MHB Limit involving a hyperbolic function

    Hello all, I am trying to solve a limit: \[\lim_{x\rightarrow 0}\frac{sinh (x)}{x}\] I found many suggestions online, from complex numbers to Taylor approximations. Finally I found a reasonable solution, but one move there doesn't make sense to me. I am attaching a picture: I have marked...
  14. J

    Scattering amplitude in scattering from a delta function

    I tried to calculate the Fourier transform to get the amplitude, but I got lost
  15. Math Amateur

    MHB Operator Norm and Distance Function .... Browder, Proposition 8.6 ....

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  16. K

    Mathematica Finding a function to best fit a curve

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  17. hilbert2

    A Function differentiability and diffusion

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  18. Y

    MHB Limit of integer part function using Sandwich rule

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  19. berkeman

    Medical Stem Cell Injections Improve Motor, Sensory Function Post Spinal Cord Injury

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  20. Z

    I Mobility and resistivity as a function of temperature

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  21. V

    MHB Function Help - Rolle's Theorem or the Mean Value Theorem?

    Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0. (a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0. (b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0. If you use a major theorem, then cite the theorem...
  22. D

    I The domain of a multivariable function

    hey there I'm struggling on finding the domain of the following function log (xy2)+x2y) I then do xy(y+x)>0 but then i don't know what to do with xy one attempt \begin{cases} y+x>0\\ x>0\\ y>0 \end{cases} union \begin{cases} y+x<0\\ x<0\\ y<0 \end{cases} but this doesn't lead to the...
  23. G

    MHB When the function is not constant

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  24. giveortake

    Engineering Dirac Delta Function in an Ordinary Differential Equation

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  25. S

    I Operator for the local average of a growing oscillating function

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  26. L

    Minimizing as a function of variables

    As promised, here is the original question, with the integral written in a more legible form.
  27. R

    Engineering Find Vout as a function of V1 and V2 (OP Amps)

    Here's the circuit in question: Solution: Now, when I try simulating in LTSpice, this is what I get: So, Vout appears to be around -13 V, which doesn't agree with the equation if V1=V2= 5 is plugged in. Does anyone see the mistake here? THanks.
  28. Saracen Rue

    I Area enclosed by a function involving 2 power towers

    I've been playing around with Up-Arrow notation quite a lot lately and have come up with the following "thought experiment" so to speak. Consider the following function: $$f(x)=(−ln(x↑↑(2k)))↑↑(2k+1)$$ $$\text{Where }k∈\mathbb{Z} ^+$$ In the image below we can see some examples of what this...
  29. jk22

    I Integration : Are a function and it's derivative independent?

    The question is a bit confused, but it refers to if the following integration is correct : $$I=\int \frac{1}{1+f'(x)}f'(x)dx$$ $$df=f'(x)dx$$ $$\Rightarrow I=\int\frac{1}{1+f'}df=?\frac{f}{1+f'}+C$$ The last equality would come if I suppose $f,f'$ are independent variables.
  30. Schwann

    B Can PDF values be equal to zero at some given points?

    Suppose we have a function which looks like this: It seems like it meets criteria of probability density functions: this function is asymptotic to zero as x approaches infinity and also it is not negative. My question is: if at some points this function reaches zero (as I have shown above)...
  31. H

    I Polynomial approximation of a more complicated function

    There is an arbitrarily complicated function F(x,y,z). I want to find a simpler surface function G(x,y,z) which approximates F(x,y,z) within a region close to the point (x0,y0,z0). Can I write a second-order accurate equation for G if I know F(x0,y0,z0) and can compute the derivatives at the...
  32. F

    How to derive a log-antilog opamp square law transfer function?

    Firstly, this is not a homework question. I found a worksheet online with an example of a square law circuit built using log-antilog operational amplifiers. I tried to derive the transfer function but I can't seem to eliminate the reverse saturation current term ##I_S##. I would really...
  33. Wrichik Basu

    Python RecursionError at a place where I have defined no recursive function

    Here is the code that I wrote: import numpy as np global m, n, p, q, arr1, arr2def input(): # Input for first matrix: print("Enter the number of rows of the first matrix: ", end="") globals()['m'] = int(input()) print("Enter the number of columns of the first matrix: ", end="")...
  34. D

    Odd/even function and critical points

    I have ##3x^{2/3}## as an even function although there is some debate as to this in another thread I started but the (5-x) factor means the function is neither odd or even. I also see the domain as all real numbers. Hopefully this is right ? To find the critical points I differentiate f(x) to...
  35. M

    Show that the given Green Function is the propagator of a certain Lagrangian

    My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from ##\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle##, which in the end is a transition amplitude and can be interpreted as...
  36. Math Amateur

    MHB Complex Function Theory: Explaining Example 1.5, Section 1.2, Chapter III

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I have yet another question regarding Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...
  37. RicardoMP

    A What is a dressing function?

    Consider, for example, the gluon propagator $$D^{\mu\nu}(q)=-\frac{i}{q^2+i\epsilon}[D(q^2)T^{\mu\nu}_q+\xi L^{\mu\nu}_q]$$ What exactly is the renormalized gluon dressing function ##D(q^2)## and what is its definition? My interest is in knowing if I can then write the bare version of this...
  38. Math Amateur

    MHB Complex Square Root Function: Qs from Bruce P. Palka's Ex. 1.5, Ch. III

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need further help with other aspects of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...
  39. Math Amateur

    MHB Differentiating Complex Square Root Function: Bruce P. Palka, Ex. 1.5, Ch. III

    I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need help with an aspect of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III, reads as...
  40. Wi_N

    If f''(x)=0, how do you find the convexity of the function?

    so ergo the function is neither convex nor concave. but graphing it in a machine it looks convex...
  41. S

    Scrambler function question

    Hi, in some standards such as JESD204B or DVB-S2 a so called scrambler function is defined. As far as I understand this scrambler is a means of spreading spectrum but in data link layer. Is it correct? Senmeis
  42. arcTomato

    I How to derive the Fourier transform of a comb function

    Dear all. I'm learning about the discrete Fourier transform. ##I(\nu) \equiv \int_{-\infty}^{\infty} i(t) e^{2 \pi \nu i t} d t=\frac{N}{T} \sum_{\ell=-\infty}^{\infty} \delta\left(\nu-\ell \frac{N}{T}\right)## this ##i(t)## is comb function ##i(t)=\sum_{k=-\infty}^{\infty}...
  43. K

    Piecewise Function: Intervals of Increase/Decrease & Extremes/Asymptotes

    a) At which intervals are f strictly increasing and at what intervals are f strictly decreasing. Should I just find the derivative of both of the functions? If so, I get that the function is increasing at the intervals (−∞,0) and (0,∞). Is this right, or can I just say that the function is...
  44. K

    Optimizing Walking Time with a Function: Finding the Best Path to KFC

    Ok. So if i sketch the curve I can see that this pound has a shape of a square. Ann and KFC has the same distance from the pond. I'm able to calculate the time for Ann to walk around the pond, and if she walks in a straight line from where she stands to KFC. If she walks around it will take...
  45. Prabs3257

    Kinetic energy as a function of time

    I got acceleration by dividing force by m then replaced a by dv/dt and then integrated it to get velocity as a fxn of time and hence got kinetic energy but problem is my ans does not match with any option can someone please compare their ans
  46. bhobba

    A Does The Use Of The Zeta Function Bypass Renormalization

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  47. megf

    I Probability Density Function of the Product of Independent Variables

    How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b? I know how to use the method to calculate it for a/b - which gives 1/pi*(a²/b²+1) - using variable substitution and the jacobian matrix and determinant, but which...
  48. S

    Determine the range of a function using parameter differentiation

    The strategy here would probably be to find a differential equation that ##f## satisfies, but differentiating with respect to ##x## using Leibniz rule yields ##f'=\int_x^{2x} (-te^{-t^2x}) \ dt + \frac{2e^{-4x^3}-e^{-x^3}}{x}## Continuing to differentiate will yield the integral term again...
  49. N

    Variance of a function which has diverging <x^2>

    I found that <x> of p(x) = 1/π(x2 + 1) is 0. But its <x^2> diverges. I don't know if there are other ways of interpreting it besides saying that the variance is infinity. I usually don't see variance being infinity, so I'm not sure if my answer is correct. So, can variance be infinity? And does...
  50. Haynes Kwon

    I Fourier Transform of the Wave function

    Given that the wave function represented in momentum space is a Fourier transform of the wave function in configuration space, is the conjugate of the wave function in p-space is the conjugate of the whole transformation integral?
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