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

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

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

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

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

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

Multiply fx * f
fx + fy = f2x * y

Multiply f2x * y
fx + fy = f2xy

Solving
fx + fy = f2xy

Solving for variable 'f'.

Combine like terms: f2xy + -1f2xy = 0
fx + fy + -1f2xy = 0

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

Simplifying
x + y + -1fxy = 0

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

Solving
-1fxy + x + y = 0

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

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

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

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

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

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

Simplifying
f = y-1 + x-1

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