Fontem paginae Functio inspicere

[[Fasciculus:Graph of example function.svg|thumb|[[graphum (mathematica)|Charta]] functionis exemplaris,<br /> <math>\begin{align}&\scriptstyle  \\ &\textstyle f(x) = \frac{(4x^3-6x^2+1)\sqrt{x+1}}{3-x}\end{align}</math><br /> <!--Both the domain and the range in the picture are the set of real numbers between −1 and 1.5.--> ]]
'''Functio''' in arte [[mathematica]] est congruentia inter duas [[copia]]s, quae ad quodque elementum primae copiae unum elementum secundae copiae destinat.<ref>Behnke et al, p. 64.</ref> Prima copia ''dominium'' dicitur,  altera ''codominium.'' Si <math>x</math> quoddam elementum primae copiae designat, <math>x</math> variabilis independens est. Si <math>f</math> functio est, quae mathematice per <math>y = f(x)</math> scribitur et significat "<math>y</math> esse elementum codominii ad elementum <math>x</math> dominii destinatum", variabilis <math>y</math> dependens appellatur.

Si plura elementa sunt, quae <math>y</math> ad elementum <math>x</math> destinari possint, congruentia ''non'' est
functio. Exempli gratia: sit <math>f(x) = \pm \sqrt{x}</math> sintque dominium et codominium copiae
[[numerus|numerorum]] realium <math>\mathbb{R}</math>. Haec congruentia non est functio, quod elemento <math>x</math> (velut 4) duo elementa (i. e. 2, -2) attribuuntur. Sin autem codominium est copia numerorum realium non-negativorum vel functio est velut <math>f(x) = + \sqrt{x}</math>, haec congruentia functio appellatur.

Functionem definire licet per formulam aut regulam aut tabulam, dum modo unum elementum codominii sit quod ad quodque elementum dominii destinetur.  Functiones sunt species [[relatio (mathematica)|relationis]]:  functio <math>f</math> est [[copia]] [[par ordinatum|parium ordinatarum]] (a, b) ut <math>f(a) = b</math>.

[[Analysis mathematica|Analysis]] est theoria functionum. Analysis [[numerus realis|numerorum realium]] est theoria functionum quarum dominium (et codominium) est <math>\mathbb{R}</math>; [[numerus complexus|analysis numerorum complexorum]] est analysis earum, quarum dominium est <math>\mathbb{C}</math>. G. H. Hardy dicit, "Illa notio, quantitatem variabilis ex quadam alia dependere, fortasse notio potissima in tota arte mathematica est."<ref>Hardy, p. 40.</ref>

Si dominium est copia quantitatum binarum, sicut <math>\mathbb{R}^2</math>, functio duas variabiles independentes habet. Exempli gratia, <math>f(x, y) = x^2 + y^2</math>. Quae functio <math>f</math> par elementorum <math>(x, y)</math> ad unum elementum codominii (quod est <math>\mathbb{R}</math>) destinat, velut par <math>(2, 3)</math> ad <math>2^2 + 3^2 = 4 + 9 = 13</math>. Functiones autem tres, quattuor pluresve variabiles independentes habere possunt.

Altera notatio functionum est ''notatio lambda,'' quae variabiles independentes post lambda litteram enumerat. Scribitur: <math>f = \lambda(x).x^2</math> quod eandem functionem atque <math>f(x) = x^2</math> describit. Forma sicut <math>\lambda(x).x^2</math> appellatur [[combinator|combinatoria]].

Si ad quoddam elementum <math>y</math> codominii aut nullum aut unum elementum <math>x</math> dominii destinatur, functio appellatur [[functio iniectiva]], aut functio unum elementum uni elemento attribuens. Si omne elementum <math>y</math> codominii habet elementum <math>x</math> (aut plura elementa <math>x_1, x_2, x_3,</math> ...) dominii quod ad <math>y</math> destinatur, functio appellatur [[functio superiectiva]]. Functio quae simul iniectiva et superiectiva est, [[functio biiectiva]] appellatur.

Quaedam functio biiectiva <math>f</math> habet [[functio inversa|functionem inversam]] <math>f^{-1}</math>, cuius dominium est codominium functionis <math>f</math> cuiusque codominium dominium functionis <math>f</math>. Si <math>f(x) = y</math>, tum est <math>f^{-1}(y) = x</math>. Exempli gratia: functio <math>f(x) = x/2</math> habet functionem inversam <math>f^{-1}(x) = 2x</math>. Formulam functionis inversae describere saepenumero haud facile est.

''Compositio'' functionum est nova functio per quam elementum dominii primae functionis cum elemento codominii secundae functionis congruit. Si <math>y = f(x), y = g(x)</math> sunt functiones, et si dominium functionis <math>f</math> est (aut continet) codominium functionis <math>g</math>, scribi potest <math>f \circ g = f(g(x))</math>. Exempli gratia, sint <math>f(x) = x^2, g(x) = \sin(x)</math>. Nunc <math>f \circ g = f(g(x)) = (\sin(x))^2</math>, et <math>g \circ f = g(f(x)) = \sin(x^2)</math>. Non sunt eaedem functiones: si <math>x = \pi, f(g(x)) = (\sin(\pi))^2 = 0</math>, sed <math>g(f(x)) = \sin(\pi^2) \approx -0.43</math>.

Copia omnium functionum invertibilium quarum dominium et codominium eadem copia est [[caterva (mathematica)|caterva]] appellatur. Idem factor catervae est functio quae quodque elementum cum eodum elemento coniungit: <math>f(x) = x</math>; operatio catervae est compositio.

== Notae ==
<references/>

{{NexInt}}
*[[Functio biiectiva]]
*[[Functio differentiabilis]]
*[[Functio exponentialis]]
*[[Functio linearis]]
*[[Graphum (mathematica)]]
*[[Polynomium]]
*[[Programmatura functionalis]]
*[[Signum (functio)]]

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