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. H

    Partition function for a spin i

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  2. Jimyoung

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  3. S

    Comp Sci Plot periodic function with Fourier coefficients

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  4. J

    B Solving Kepler's 1st law as a function of time

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  5. Q

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  6. P

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  7. chwala

    Solve this problem that involves a probability density function

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  8. K

    A How to check if a function doesn't depend on a variable?

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  9. T

    I Closed Form for Complex Gamma Function

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  10. G

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  11. PhysicsRock

    I Is the solution to this problem as trivial as I think?

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  12. H

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  13. Barbequeman

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  14. H

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  15. S

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  16. LCSphysicist

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  17. N

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  18. chwala

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  19. N

    I Why can't we just integrate a simple function?

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  20. PhysicsRock

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  21. Euge

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  22. chwala

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  23. Franklie001

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  24. A

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  25. V

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  26. H

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  27. T

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  28. H

    Calculating total derivative of multivariable function

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  29. H

    Finding roots: cosine function of x

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  30. chwala

    Solve the problem involving probability density function

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  31. A

    Engineering How do radial tooth clutches function?

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  32. chwala

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  33. cnh1995

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  34. topsquark

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  35. J

    Using Gauss's Law to Calculate Charge Density Function

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  36. besebenomo

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  37. Ennio

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  38. M

    Python Any help with this error in Scipy minimize function?

    Hi PF! When I execute the code below: import numpy as np from scipy.stats import t import scipy.optimize as optimizeglobal data data = np.random.normal(loc=50, scale=1, size=(2400, 1)).flatten() def L(F): M = 250 lmda = 0.97 sig_0 = F[0] for i in range(1, 12): sig_0 +=...
  39. Ennio

    A Calculate the exact or approximate length of one period of the wave function

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  40. M

    Python Scipy minimizer function and function executions

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  41. The Bill

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  42. J

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  43. jjson775

    I Understanding Wave Function Probabilities for a Free Particle

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  44. ergospherical

    I 4D d'Alembert Green's function for linearised metric

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  45. yklee

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  46. S

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  47. patric44

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  48. MevsEinstein

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  49. H

    I Partition function for continuous spectrum

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  50. Aethermimicus

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