One thing that might be useful to know is that the summation property of this function does not have to be restricted to two terms. For example, f(a^2 + b^2 + c^2) = (f(a))^2 + (f(b))^2 + (f(c))^2, which is easily proven through iteration of the sum property.

The reason why this property is useful is that, although not every number is a sum of two perfect squares, every number must be a sum of some finite number of perfect squares (at worst, n is the sum of n ones).