Composition/Polynomial/Example

The composition of

$F\colon \mathbb {R} \longrightarrow \mathbb {R} ,t\longmapsto t^{3},$

and

$G\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}-x,$

is given by

${}(G\circ F)(t)=(t^{3})^{2}-t^{3}=t^{6}-t^{3}\,.$

However,

${}(F\circ G)(x)=(x^{2}-x)^{3}=x^{6}-3x^{5}+3x^{4}-x^{3}\,.$

Hence, the composition of two mappings depends on the ordering.