Compositio (mathematica)

Compositio duarum functionum ${\displaystyle f:X\rightarrow Y,x\mapsto f(x)=y}$ et ${\displaystyle g:Y\rightarrow Z,y\mapsto g(y)}$ est functio ${\displaystyle h:X\rightarrow Z,}$

${\displaystyle x\mapsto g(y)=g(f(x)).}$

${\displaystyle g\circ f}$ pro h scribi solet.

Exempla

• Sit ${\displaystyle f(x)=2x^{2}-4x+3,x\in \mathbb {R,} }$ et ${\displaystyle g(y)=\cos(y),y\in W_{f}\subset \mathbb {R} ;}$ tum ${\displaystyle (g\circ f)(x)=\cos(2x^{2}-4x+3),x\in \mathbb {R} .}$

• Sit ${\displaystyle f:\mathbb {R_{+} \rightarrow \mathbb {R_{+} ,x\mapsto x^{2}$ et ${\displaystyle g:\mathbb {R_{+} \rightarrow \mathbb {R_{+} ,y\mapsto {\sqrt {y};}$ tum
  ${\displaystyle (g\circ f)(x)={\sqrt {x^{2}=x}$ et ${\displaystyle (f\circ g)(y)={({\sqrt {y})}^{2}=y.}$
  Compositiones, quae aequationi ${\displaystyle f\circ g=\mathrm {id} _{Y}$ et ${\displaystyle g\circ f=\mathrm {id} _{X}$ sufficiant, inversae appellantur.