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

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


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

Multiply f * x
fx + -1fy = fx * f * y

Multiply fx * f
fx + -1fy = f2x * y

Multiply f2x * y
fx + -1fy = f2xy

Solving
fx + -1fy = f2xy

Solving for variable 'f'.

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

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

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

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

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

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

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

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

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

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

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

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