# Functions of several variables composition

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?

• Maybe you want @* ? Jul 24, 2020 at 19:04
• ^ 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]]] Jul 24, 2020 at 19:37