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. chwala

    Write the given hyperbolic function as simply as possible

    My take; ##2\cosh x = e^x +e^{-x}## I noted that i could multiply both sides by ##e^x## i.e ##e^x⋅2\cosh x = e^x(e^x +e^{-x})## ##e^x⋅2\cosh x = e^{2x}+1## thus, ##\dfrac{e^x}{1+e^{2x}}=\dfrac{\cosh x + \sinh x}{e^x⋅2\cosh x}## ##= \dfrac{\cosh x +...
  2. C

    Is ##f(x)=2^{x}-1## considered an exponential function?

    I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am...
  3. ananonanunes

    Find limit of multi variable function

    This is what I did: $$\lim_ {(x,y) \rightarrow (1,0)} {\frac {g(x)(x-1)^2y}{2(x-1)^4+y^2}}=\lim_ {(x,y) \rightarrow (1,0)} {g(x)y\frac {(x-1)^2}{2(x-1)^4+y^2}}$$ I know that ##\lim_ {(x,y) \rightarrow (1,0)} {g(x)y}=0## and that ##\frac {(x-1)^2}{2(x-1)^4+y^2}## is limited because ##0\leq...
  4. C

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  5. Like Tony Stark

    Mixed states and total wave function for three-Fermion-systems

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  6. G

    I Independence of Trace-Partition function

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  7. R

    Expressing Feynman Green's function as a 4-momentum integral

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  8. M

    B How do I invert this exponential function?

    Preface: I have not done serious math in years. Today I tried to do something fancy for a game mechanic I'm designing. I've got an item with a variable power level. It uses x amount of ammo to produce f(x) amount of kaboom. Initially it was linear, e.g. fL(x) = x, but I didn't like the scaling...
  9. K

    I Best fit to an oscillating function

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  10. B

    I Limit as a function, not a value

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  11. C

    Finding the domain of a composite function

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  12. E

    I Integration of Bessel function products (J_1(x)^2/xdx)

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  13. S

    I Geometry of series terms of the Riemann Zeta Function

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  14. E

    I Determine ## \beta ## as a function of ##\theta## linkage

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  15. C

    Limit of a rational function with a constant c

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  16. O

    Symbolic integration of a Bessel function with a complex argument

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  17. C

    I How to calculate the Fermi surface nesting function

  18. vinicius_linhares

    Analysis of the Ground Function: f(x) with $$f''(\bar{x})=0$$

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  19. S

    Trying to reconcile function composition problems with sets & formulas

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  20. F

    Looking for a particular function

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  21. S

    I Load on leaf blower as function of outlet shape

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  22. E

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  23. Euge

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  24. abdulbadii

    Failed function within SMPS giving weak current

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  25. loversphisics

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  26. C

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  27. M

    Engineering How to implement a transfer function in Simulink with variable coefficients?

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  28. yucheng

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  29. I

    I QED/Quantum Mechanics: Probability or Spatial Function?

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  30. L

    I Create a surjective function from [0,1]^n→S^n

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  31. C

    Formula for Xo and Yo for graph of quadratic function

    Can someone please tell me where I am wrong. I am learning how to write ##a^{2} + bx + c## in this form ##f(x)= a(X-X_0)^{2} +Y_0##. The method used in my textbook is a reduction to the perfect square. And it goes like this: ##f(x)=ax^2+bx+c## ##=a[x^2+\frac{b}{a}x]+c## ##=a\left [...
  32. R

    I Multivariable function optimization inconsistency

    Mentor note: For LaTeX here at this site, don't use single $ characters -- they don't work at all. See our LaTeX tutorial from the link at the lower left corner of the input text pane. I have a function dependent on 4 variables ##f(r_1,r_2,q_1,q)##. I'm looking to minimize this function in the...
  33. V

    Is it possible to find the range of this function?

    I can get the domain, but getting the range seems impossible. Domain $$x-5=0$$ $$x =5$$ $$\therefore x \in (- \infty ,5) \cup (5, + \infty)$$ Range  I can simplify the function to the form below, but I don't know how to go from there. $$ f(x)= x + 5 + \frac {1}{x-5}$$  
  34. P

    A Norm 2, f Integrable function, show: ##||f-g||_2<\epsilon##

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  35. hagopbul

    An idea to improve kidney function

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  36. C

    The derivative of uv wrt x using st function (homework problem)

    TL;DR Summary: I attempt to find the derivative of uv with respect to x using non standard analysis, hyperreals, and the standard part function st; I take u to be a function of x, and I also take v to be a function of x. Hello everyone! I've been learning about non standard analysis concepts...
  37. P

    A Vector calculus - Prove a function is not differentiable at (0,0)

    ##f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}## I showed it partial derivatives exist at ##(0,0)##, also it is continuous as ##(0,0)## but now I have to show if it differentiable or not at ##(0,0)##. According to answers it is not...
  38. H

    B Can we all agree "consciousness" is not required to collapse wave function?

    I see this written or talked about so often. Pop-sci for sure. But, whatever the wave function is, and whatever might collapse it, can we agree consciousness is not required to collapse it? I.E., the moon was there before "conscious" beings, on this planet or elsewhere, viewed it? Is at...
  39. H

    B Why does anyone think gravity might collapse wave function?

    Why on Earth does anyone, let along Roger Penrose, think gravity might be what causes the wave function to collapse? The most basic experiment in quantum physics, the double slit experiment, shows that collapse is most closely analogous to whether or not the item at issue (for example, an...
  40. PhysicsRock

    How to Show Linearity of a Function?

    I don't really know how I am supposed to approach that. In general, I know how to show that a function is linear, which is to show that ##f(\alpha \cdot x) = \alpha \cdot f(x)## and ##f(x_1 + x_2) = f(x_1) + f(x_2)##. However, for this specific function, I have no idea, since there is nothing...
  41. mcastillo356

    Why does this function make it easy to prove continuity with sequences?

    I've been given the proof, but don't understand; to calculate the limit of ##f## when ##x## tends to zero it's enough to see that if ##\{x_n\}_{n=1}^\infty## is a sequence that tends to ##0##, then...
  42. C

    Deriving position function for object in SHM

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  43. P

    I Can an inverse function of a special cubic function be found?

    Like this one: ##f(x)=3x^3 -18x^2 +36x##.Is there any way to find the inverse of this function without using the general solution to cubic equation?
  44. G

    Apply the Legendre Transformation to the Entropy S as a function of E

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  45. V

    Matlab Function for Composite Simpson

    function I=main_simpson(a,b,tol) f = @(x) sin(1./x); SO = 0; N = 10; S = 1; while (abs(S-SO)>tol) SO = S; h = (b-a)/(2*N); i = 0:N-1; xi = a+2*i*h; xi1 = a+2*(i+0.5)*h; xi2 = a+2*(i+1)*h; S = (h/3)*sum(f(xi)+4*f(xi1)+f(xi2)); N = 2*N; end end <Moderator's note...
  46. Expiring

    I Showing That a Function Does Not Have Two Distinct Roots

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  47. jaumzaum

    B Can a function have two fundamental periods?

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  48. chwala

    Prove the hyperbolic function corresponding to the given trigonometric function

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  49. K

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  50. T

    I Hubble Parameter as function of time in universe models

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