f(x)xf(y)=f(x+y+xy)

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


Simplifying
f(x) * xf(y) = f(x + y + xy)

Multiply f * x
fx * fx * y = f(x + y + xy)

Multiply fx * fx
f2x2 * y = f(x + y + xy)

Multiply f2x2 * y
f2x2y = f(x + y + xy)

Reorder the terms:
f2x2y = f(x + xy + y)
f2x2y = (x * f + xy * f + y * f)
f2x2y = (fx + fxy + fy)

Solving
f2x2y = fx + fxy + fy

Solving for variable 'f'.

Reorder the terms:
-1fx + -1fxy + -1fy + f2x2y = fx + fxy + fy + -1fx + -1fxy + -1fy

Reorder the terms:
-1fx + -1fxy + -1fy + f2x2y = fx + -1fx + fxy + -1fxy + fy + -1fy

Combine like terms: fx + -1fx = 0
-1fx + -1fxy + -1fy + f2x2y = 0 + fxy + -1fxy + fy + -1fy
-1fx + -1fxy + -1fy + f2x2y = fxy + -1fxy + fy + -1fy

Combine like terms: fxy + -1fxy = 0
-1fx + -1fxy + -1fy + f2x2y = 0 + fy + -1fy
-1fx + -1fxy + -1fy + f2x2y = fy + -1fy

Combine like terms: fy + -1fy = 0
-1fx + -1fxy + -1fy + f2x2y = 0

Factor out the Greatest Common Factor (GCF), 'f'.
f(-1x + -1xy + -1y + fx2y) = 0


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

Simplifying
f = 0

Solving
f = 0

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

Simplifying
f = 0

Subproblem 2
Set the factor '(-1x + -1xy + -1y + fx2y)' equal to zero and attempt to solve:

Simplifying
-1x + -1xy + -1y + fx2y = 0

Reorder the terms:
fx2y + -1x + -1xy + -1y = 0

Solving
fx2y + -1x + -1xy + -1y = 0

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

Add 'x' to each side of the equation.
fx2y + -1x + -1xy + x + -1y = 0 + x

Reorder the terms:
fx2y + -1x + x + -1xy + -1y = 0 + x

Combine like terms: -1x + x = 0
fx2y + 0 + -1xy + -1y = 0 + x
fx2y + -1xy + -1y = 0 + x
Remove the zero:
fx2y + -1xy + -1y = x

Add 'xy' to each side of the equation.
fx2y + -1xy + xy + -1y = x + xy

Combine like terms: -1xy + xy = 0
fx2y + 0 + -1y = x + xy
fx2y + -1y = x + xy

Add 'y' to each side of the equation.
fx2y + -1y + y = x + xy + y

Combine like terms: -1y + y = 0
fx2y + 0 = x + xy + y
fx2y = x + xy + y

Divide each side by 'x2y'.
f = x-1y-1 + x-1 + x-2

Simplifying
f = x-1y-1 + x-1 + x-2

Reorder the terms:
f = x-2 + x-1 + x-1y-1
Solution
f = {0, x-2 + x-1 + x-1y-1}

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