# Functions of several variables composition

Mathematica Asked on January 2, 2022

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

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

How could this process be programmed in Mathematica12?

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