iterated function permutation with multiple inputs

Question

For any given function f that takes two inputs, x and y, and for any arbitrary amount of iterations, where both x and y could be any arbitrary real number would the ordering of possible y inputs be commutative? For example can I prove

f(f(f(x, n₀), n₁), n₂)

would always be equal to

f(f(f(x, n₁), n₂), n₀)

for any arbitrary amount of n values and for any arbitrary ordering of those n values?

Answer 1

It's not true. For example, consider $f(x,y) = \sqrt{x+y}$. Then you can check that $$ \sqrt{\sqrt{3}+3} = f(f(1,2),3) \neq f(f(1,3),2) = 2. $$