f(x*f(y)+y*f(x)-x*y)=f(x)*f(y)

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

Simplifying
f(x * f(y) + y * f(x) + -1x * y) = f(x) * f(y)

Multiply x * f
f(fx * y + y * f(x) + -1x * y) = f(x) * f(y)

Multiply fx * y
f(fxy + y * f(x) + -1x * y) = f(x) * f(y)

Multiply y * f
f(fxy + fy * x + -1x * y) = f(x) * f(y)

Multiply fy * x
f(fxy + fxy + -1x * y) = f(x) * f(y)

Multiply x * y
f(fxy + fxy + -1xy) = f(x) * f(y)

Combine like terms: fxy + fxy = 2fxy
f(2fxy + -1xy) = f(x) * f(y)
(2fxy * f + -1xy * f) = f(x) * f(y)

Reorder the terms:
(-1fxy + 2f2xy) = f(x) * f(y)
(-1fxy + 2f2xy) = f(x) * f(y)

Multiply f * x
-1fxy + 2f2xy = fx * f * y

Multiply fx * f
-1fxy + 2f2xy = f2x * y

Multiply f2x * y
-1fxy + 2f2xy = f2xy

Solving
-1fxy + 2f2xy = f2xy

Solving for variable 'f'.

Combine like terms: 2f2xy + -1f2xy = 1f2xy
-1fxy + 1f2xy = f2xy + -1f2xy

Combine like terms: f2xy + -1f2xy = 0
-1fxy + 1f2xy = 0

Factor out the Greatest Common Factor (GCF), 'fxy'.
fxy(-1 + f) = 0

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

Simplifying
fxy = 0

Solving
fxy = 0

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

Simplifying
fxy = 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 '(-1 + f)' equal to zero and attempt to solve:

Simplifying
-1 + f = 0

Solving
-1 + f = 0

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

Add '1' to each side of the equation.
-1 + 1 + f = 0 + 1

Combine like terms: -1 + 1 = 0
0 + f = 0 + 1
f = 0 + 1

Combine like terms: 0 + 1 = 1
f = 1

Simplifying
f = 1
Solution
f = {1}