function construction - Composition of mappings not working as expected

I have two functions $f,g:\mathbb{R}^2 \to \mathbb{R}^2$ and I define a third one $h:\mathbb{R}^2 \to \mathbb{R}^2$ as the composition $$h(x,y) = g(f(x,y))$$ I'm trying to get this function into Mathematica as a composition because I still want the functions $f,g$ and I don't want to have to manually update $h$ when I change $f$ or $g$.

So here is what I'm currently doing :

1. Define f[x_, y_] := {E^x + y, Sin[2 x]}
2. Define g[x_, y_] := {2 x + Cos[y], E^(x + y)}
3. Try the obvious definition of h[x_, y_] := g[f[x, y]].

This doesn't seem to work. I'll just get things like h[0, 0] returning g[{1, 0}]. I could write $h$ using the projection of $f$ onto the standard basis for $\mathbb{R}^2$ with the Projection function but this seems needlessly cumbersome and adds conjugate statements everywhere. Can anybody point out a better direction for defining $h$ as a composition?

Also I have tried the Mathematica function Composition and it behaved the same way.

Answer

Your functions take two arguments as input. Yet they return one argument as output. So the output of f cannot serve as the input of g and vice versa. You can't compose the functions with until the outputs of one function can serve as the input for the other function. Reduce your arguments in f, g by placing them in a list.

This appears to work:

f[{x_, y_}] := {E^x + y, Sin[2 x]}

g[{x_, y_}] := {2 x + Cos[y], E^(x + y)}
h[{x_, y_}] := g[f[{x, y}]]

Example

g[f[{2, 3}]]
f[g[{2, 3}]]