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

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


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

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

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

Multiply f * y
fxy = fx + fy

Solving
fxy = fx + fy

Solving for variable 'f'.

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

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

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

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

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

Factor out the Greatest Common Factor (GCF), 'f'.
f(-1x + xy + -1y) = 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 + xy + -1y)' equal to zero and attempt to solve:

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

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

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

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

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

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

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

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

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

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

Simplifying
0 = x + -1xy + y

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
f = {0}

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