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.

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.

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

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