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

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

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

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

Multiply fx * f
f2x * y = f(x + y)

Multiply f2x * y
f2xy = f(x + y)
f2xy = (x * f + y * f)
f2xy = (fx + fy)

Solving
f2xy = fx + fy

Solving for variable 'f'.

Reorder the terms:
-1fx + -1fy + f2xy = fx + fy + -1fx + -1fy

Reorder the terms:
-1fx + -1fy + f2xy = fx + -1fx + fy + -1fy

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

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

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

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

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

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

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

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

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

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

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

Divide each side by 'xy'.
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}