Functio

Functio in arte mathematica est congruentia inter duas copias, quae ad quodque elementum primae copiae unum elementum secundae copiae destinat. Prima copia dominium dicitur,  altera codominium. Si $x$ quoddam elementum primae copiae designat, $x$ variabilis independens est. Si $f$ functio est, quae mathematice per $y=f(x)$ scribitur et significat "$y$ esse elementum codominii ad elementum $x$ dominii destinatum", variabilis $y$ dependens appellatur.

Charta functionis exemplaris,
${\begin{aligned}&\scriptstyle \\&\textstyle f(x)={\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}$

Si plura elementa sunt, quae $y$ ad elementum $x$ destinari possint, congruentia non est functio. Exempli gratia: sit $f(x)=\pm {\sqrt {x}}$ sintque dominium et codominium copiae numerorum realium $\mathbb {R}$. Haec congruentia non est functio, quod elemento $x$ (velut 4) duo elementa (i. e. 2, -2) attribuuntur. Sin autem codominium est copia numerorum realium non-negativorum vel functio est velut $f(x)=+{\sqrt {x}}$, 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 relationis: functio $f$ est copia parium ordinatarum (a, b) ut $f(a)=b$.

Analysis est theoria functionum. Analysis numerorum realium est theoria functionum quarum dominium (et codominium) est $\mathbb {R}$; analysis numerorum complexorum est analysis earum, quarum dominium est $\mathbb {C}$. G. H. Hardy dicit, "Illa notio, quantitatem variabilis ex quadam alia dependere, fortasse notio potissima in tota arte mathematica est."

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

Altera notatio functionum est notatio lambda, quae variabiles independentes post lambda litteram enumerat. Scribitur: $f=\lambda (x).x^{2}$ quod eandem functionem atque $f(x)=x^{2}$ describit. Forma sicut $\lambda (x).x^{2}$ appellatur combinatoria.

Si ad quoddam elementum $y$ codominii aut nullum aut unum elementum $x$ dominii destinatur, functio appellatur functio iniectiva, aut functio unum elementum uni elemento attribuens. Si omne elementum $y$ codominii habet elementum $x$ (aut plura elementa $x_{1},x_{2},x_{3},$ ...) dominii quod ad $y$ destinatur, functio appellatur functio superiectiva. Functio quae simul iniectiva et superiectiva est, functio biiectiva appellatur.

Quaedam functio biiectiva $f$ habet functionem inversam $f^{-1}$, cuius dominium est codominium functionis $f$ cuiusque codominium dominium functionis $f$. Si $f(x)=y$, tum est $f^{-1}(y)=x$. Exempli gratia: functio $f(x)=x/2$ habet functionem inversam $f^{-1}(x)=2x$. 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 $y=f(x),y=g(x)$ sunt functiones, et si dominium functionis $f$ est (aut continet) codominium functionis $g$, scribi potest $f\circ g=f(g(x))$. Exempli gratia, sint $f(x)=x^{2},g(x)=\sin(x)$. Nunc $f\circ g=f(g(x))=(\sin(x))^{2}$, et $g\circ f=g(f(x))=\sin(x^{2})$. Non sunt eaedem functiones: si $x=\pi ,f(g(x))=(\sin(\pi ))^{2}=0$, sed $g(f(x))=\sin(\pi ^{2})\approx -0.43$.

Copia omnium functionum invertibilium quarum dominium et codominium eadem copia est caterva appellatur. Idem factor catervae est functio quae quodque elementum cum eodum elemento coniungit: $f(x)=x$; operatio catervae est compositio.

Nexus interni

• Functio biiectiva
• Functio differentiabilis
• Functio exponentialis
• Functio linearis
• Graphum (mathematica)
• Polynomium
• Programmatura functionalis

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