Mean functions

Let x = a + b and y = a - b

Then f( x , y ) = f ( a + b , a - b ) = a * b

Then f ( y , x ) = f (a -b , a + b) = f ( a + (-b) , a - (- b) ) = a * (- b) = - (a*b)

arithmetic mean = sum divided by 2

a b + (- a b) = 0

0 / 2 = 0

The above functional equation can be written as:

$f(x+y, x-y) = xy = \frac{1}{4} \cdot [(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)] = \frac{1}{4} \cdot [(x+y)^2 - (x-y)^2],$

Hence, our required function is $f(x,y) = \frac{x^2 - y^2}{4}.$ The arithmetic mean of $f(x,y)$ and $f(y,x)$ computes to:

$M = \frac{1}{2} \cdot [\frac{x^2-y^2}{4} + \frac{y^2-x^2}{4}] = \frac{1}{8} \cdot 0 = \boxed{0}.$