f(x+f(y))=f(x+y)+f(y)

Solution for f(x+f(y))=f(x+y)+f(y) equation:

Simplifying
f(x + f(y)) = f(x + y) + f(y)

Multiply f * y
f(x + fy) = f(x + y) + f(y)

Reorder the terms:
f(fy + x) = f(x + y) + f(y)
(fy * f + x * f) = f(x + y) + f(y)

Reorder the terms:
(fx + f2y) = f(x + y) + f(y)
(fx + f2y) = f(x + y) + f(y)
fx + f2y = (x * f + y * f) + f(y)
fx + f2y = (fx + fy) + f(y)

Multiply f * y
fx + f2y = fx + fy + fy

Combine like terms: fy + fy = 2fy
fx + f2y = fx + 2fy

Add '-1fx' to each side of the equation.
fx + -1fx + f2y = fx + -1fx + 2fy

Combine like terms: fx + -1fx = 0
0 + f2y = fx + -1fx + 2fy
f2y = fx + -1fx + 2fy

Combine like terms: fx + -1fx = 0
f2y = 0 + 2fy
f2y = 2fy

Solving
f2y = 2fy

Solving for variable 'f'.

Reorder the terms:
-2fy + f2y = 2fy + -2fy

Combine like terms: 2fy + -2fy = 0
-2fy + f2y = 0

Factor out the Greatest Common Factor (GCF), 'fy'.
fy(-2 + f) = 0

Subproblem 1
Set the factor 'fy' equal to zero and attempt to solve:

Simplifying
fy = 0

Solving
fy = 0

Move all terms containing f to the left, all other terms to the right.

Simplifying
fy = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(-2 + f)' equal to zero and attempt to solve:

Simplifying
-2 + f = 0

Solving
-2 + f = 0

Move all terms containing f to the left, all other terms to the right.

Add '2' to each side of the equation.
-2 + 2 + f = 0 + 2

Combine like terms: -2 + 2 = 0
0 + f = 0 + 2
f = 0 + 2

Combine like terms: 0 + 2 = 2
f = 2

Simplifying
f = 2
Solution
f = {2}