Functio

Functio in arte mathematica est congruentia inter duas copias, quae ad quodque elementum primae copiae unum elementum secundae copiae destinat.^[1] Prima copia dominium dicitur,  altera codominium. Si ${\displaystyle x}$ quoddam elementum primae copiae designat, ${\displaystyle x}$ variabilis independens est. Si ${\displaystyle f}$ functio est, quae mathematice per ${\displaystyle y=f(x)}$ scribitur et significat "${\displaystyle y}$ esse elementum codominii ad elementum ${\displaystyle x}$ dominii destinatum", variabilis ${\displaystyle y}$ dependens appellatur.

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

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

Analysis est theoria functionum. Analysis numerorum realium est theoria functionum quarum dominium (et codominium) est ${\displaystyle \mathbb {R} }$; analysis numerorum complexorum est analysis earum, quarum dominium est ${\displaystyle \mathbb {C} }$. G. H. Hardy dicit, "Illa notio, quantitatem variabilis ex quadam alia dependere, fortasse notio potissima in tota arte mathematica est."^[2]

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

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

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

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

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: ${\displaystyle f(x)=x}$; operatio catervae est compositio.

Notae

1. Behnke et al, p. 64.
2. Hardy, p. 40.

Nexus interni

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

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