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 ACT.trig.01 What is the period of the function csc 4x

    $\tiny{ACT.trig.01}$ What is the period of the function $f(x)=\csc{4x}$ $a. \pi \quad b, 2\pi \quad c. 4\pi \quad d. \dfrac{\pi}{4} \quad e. \dfrac{\pi}{2}$ well we should know the answer by observation but I had to graph it looks like $\dfrac{\pi}{2}$
  2. S

    Finding the potential function of a vector field

    Hello! So I need to find the potential function of this Vector field $$ \begin{matrix} 2xy -yz\\ x^2-xz\\ 2z-xy \end{matrix} $$ Now first I tried to check if rotation is not ,since that is mandatory for the potentialfunction to exist.For that I used the jacobi matrix,and it was not...
  3. Eclair_de_XII

    B Is the level-set for a given function unique?

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  4. chriswong

    Why Does Stoke's Stream Function Represent Fluid Flow?

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  5. H

    Confirm Limit Existence for Function f w/o Piecewise Def.

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  6. brotherbobby

    The integral of a function ##f(x)## from its graph

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  7. C

    Showing continuous function has min or max using Cauchy limit def.

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  8. C

    Using params from gsl function in integration routine

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  9. F

    I How do I do this calculation involving the SIN function?

    Hi, I am new here and hope I have posted my thread in the right forum. I have the following SIN function in Excel: =1*(SIN(2*PI()*1,6667*0,45)) The result is -1. That is what I want, so no problem. But what I want is a function that calculates the Time t value, in this example the value 0.45...
  10. evinda

    MHB How do we choose this function?

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  11. P

    I Find P(X+Y>1/2) for given joint density function

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  12. Mathman2013

    Find maximum of the function N(t) without Calculus

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  13. A

    I What is the indefinite integral of Bessel function of 1 order (first k

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  14. A

    Writing the charge density in the form of the Dirac delta function

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  15. LCSphysicist

    Find FT of Function: Solution & Explanation

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  16. N

    MHB How can we rewrite a modulus function?

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  17. C

    KarhunenLoeveDecomposition function in Mathematica

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  18. A

    Finding Absolute Minima & Maxima of a Function

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  19. jk22

    I Can Lagrange multipliers be used to find a function?

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  20. M

    MHB Piecewise function: differentiable but not continuously differentiable

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  21. greg_rack

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  22. L

    Problem with units? (trying to solve a normalized function)

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  23. boniphacy

    I Problem with function approximation

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  24. S

    I deriving a pupil function in a coherent imaging system

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  25. docnet

    B Helping a 5-Year-Old Integrate a Function in Kindergarten

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  26. anaisabel

    Grand partition function (Volume divided into N spaces)

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  27. F

    I Why is it important to convert angle units when using trigonometric functions?

    Hello, Periodic trigonometric functions, like sine and cosine, generally take an angle as input to produce an output. Functions do that: given an input they produce an output. Angles are numerically given by real numbers and can be expressed either in radians or degrees (just two different...
  28. anemone

    MHB Linear combination of sine and cosine function

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  29. Haorong Wu

    I Could this function be approximated by Dirac delta function?

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  30. Meaning

    A Is a solution of a differential equation a function of its parameters?

    Hi everyone, Imagine I have a system of linear differential equations, e.g. the Maxwell equations. Imagine my input variables are the conductivity $\sigma$. Is it correct from the mathematical point of view to say that the electric field solution, $E$, is a function of sigma in general...
  31. Lilian Sa

    Star collapse in general relativity — pressure as a function of star radius

    What I've done is using the TOV equations and I what I found at the end is: ##e^{[\frac{-8}{3}\pi G\rho]r^2+[\frac{16}{9}(G\pi\rho)^{2}]r^4}-\rho=P(r)## so I am sure that this is not right, if someone can help me knowing it I really apricate it :)
  32. MisterH

    A Fitness function for window length of filter

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  33. stevendaryl

    Insights Computing the Riemann Zeta Function Using Fourier Series

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  34. M

    Decision Theory: Discriminant function for 2D Gaussians

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  35. docnet

    Is there a smooth function in the unit ball with compact support?

    Is there a function that takes positive values only in the unit ball not including the boundary points defined by the set ##\{x^2+y^2+z^2<1\}##, and ##0## everywhere else?
  36. J

    Polymorphic conversion in return type of function

    I have an abstract class template, BinNode<E> and another class template, BSTNode<Key, E>, which is derived from BinNode<E>. Here is the relevant pieces of code from each template: template <typename E> class BinNode { public: virtual BinNode<E>* left() const = 0; virtual void...
  37. docnet

    Verify this function is a solution of the heat equation

    I spent hours looking at this and cannot figure out where the error is. I'm wondering if there is an error before the boxed expression. @Orodruin and @PeroK may I ask for your assistance?Consider a solution ##u:[0,\infty)\times \mathbb{R}^n\rightarrow \mathbb{R}## of the heat equation, ie...
  38. mcastillo356

    Thoughts on the derivative of a function

    (a) ##f(x)## is continuous only at ##x=3##: 1- If ##x\in\mathbb Q##, ##f(x)=9## at ##x=3##; around, there is ##\mathbb Q## 2- If ##x\in \mathbb R\setminus \mathbb Q##, this is the set of irrational numbers. Intuitively, if ##x## was in ##\mathbb R##, ##x^2## and ##6(x-3)+9## would meet at...
  39. N

    MHB Horizontal Asymptote of Rational Function

    Given f(x) = [sqrt{2x^2 - x + 10}]/(2x - 3), find the horizontal asymptote. Top degree does not = bottom degree. Top degree is not less than bottom degree. If top degree > bottom degree, the horizontal asymptote DNE. The problem for me is that 2x^2 lies within the radical. I can rewrite...
  40. docnet

    How to show a function is twice continuously differentiable?

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  41. Lexovix

    B Wave Function Collapse using faulty recording devices.

    Hey there! I have two questions regarding the Double Slit Experiment and the Wave Function Collapse. How effective does a measuring device have to be to cause a collapse? As in, say that every second the device has a 50% chance to turn off or on for one second, does the collapse still occur...
  42. S

    Solution of inequality of composite function involving inverse

    I can solve (i), I got x = -1.6 For (ii), I did like this: $$(f^{-1} o ~g)(x)<1$$ $$g(x)<f(1)$$ But it is wrong, the correct one should be ##g(x) > f(1)##. Why? Thanks
  43. wcjy

    Circuits - Transfer function of this LRC circuit

    Got B = 9. My school doesn't provide answers, hence could someone double check if my answers is right? Thanks!
  44. R

    Find a continuous solution to an ODE that includes a step function

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  45. icesalmon

    Engineering Minterms of a Boolean function

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  46. S

    MHB Solving the Floor Function Equation: $[x]=2x+1$

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  47. A

    A Green's function for tunneling electrons between quantum dots

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  48. C

    Finding a domain for a function

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  49. C

    MHB Finding the domain of this function.

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  50. Lord Doppler

    Engineering Transfer function of AC Transformer

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