Lambda Calculus

(◦ f) ◦ g is equivalent to λxy. g(x, f(y)).

In Haskell:

-- let h = (. f) . g, then
-- h x = ((. f) . g) x
--     = (. f) (g x)
--     = (g x) . f
-- then
-- h x y = ((g x) . f) y
--       = g x (f y)
((. (+3)) . (\x y -> x^2 + y^2)) 1 2 -- yields 26

In Mathematica:

Composition[
   CurryApplied[Composition, {2, 1}][CurryApplied[f, 1]], 
   CurryApplied[g, 2]
][x][y] (* yields g[x, f[y]] *)
 
Composition[
   CurryApplied[Composition, {2, 1}][CurryApplied[Plus, 2][3]], 
   CurryApplied[#1^2 + #2^2 &, 2]
][1][2]  (* yields 26 *)

In Racket:

#lang racket
 
(define (comp-comp f g)
  (let [[cf (curry f)]
        [cg (curry g)]]
    (compose (lambda (x) (compose x cf)) cg)))
 
(define (f x) (+ x 3))
(define (g x y) (+ (* x x) (* y y)))
 
(((comp-comp f g) 1) 2) ;; yields 1^2 + (2+3)^2 = 26

There's a package called relation-lib, the function uncurry in it makes things simpler:

#lang racket
 
(require relation/function)
 
(define (comp-comp f g)
  (let [[cf (curry f)]
        [cg (curry g)]]
    (uncurry (compose (lambda (x) (compose x cf)) cg))))
 
(define (f x) (+ x 3))
(define (g x y) (+ (* x x) (* y y)))
 
((comp-comp f g) 1 2) ;; yields 1^2 + (2+3)^2 = 26