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If we have two functions $G,F:\mathbb R^2\to\mathbb R^2$ then the mechanism $F\circ G(x,y)=F(G(x,y))$ gives us a new map $F\circ G:\mathbb R^2\to\mathbb R^2$ called its composition.

In a "schematic" view, and with two functions for an example, we would write $$ \left(\begin{array}{c}x\\y\end{array}\right) \stackrel{G}\to \left(\begin{array}{c}x^2y\\2y\end{array}\right)\qquad {\rm and}\qquad \left(\begin{array}{c}x\\y\end{array}\right) \stackrel{F}\to \left(\begin{array}{c}x-y\\x\end{array}\right) $$ then $$ \left(\begin{array}{c}x\\y\end{array}\right) \stackrel{F\circ G}\longrightarrow \left(\begin{array}{c}x^2y-2y\\x^2y\end{array}\right). $$

How could this process be programmed in Mathematica12?

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  • $\begingroup$ Maybe you want @* ? $\endgroup$ Jul 24, 2020 at 19:04
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    $\begingroup$ ^ the above is shorthand for Composition. There is also RightComposition if you want to build up the composition in order of how the functions are applied, e.g (f /* g /* h)[x] gives h[g[f[x]]] $\endgroup$
    – flinty
    Jul 24, 2020 at 19:37

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