2f(x+y)=f(x)f(y)

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Solution for 2f(x+y)=f(x)f(y) equation: 


Simplifying
2f(x + y) = f(x) * f(y)
(x * 2f + y * 2f) = f(x) * f(y)
(2fx + 2fy) = f(x) * f(y)

Multiply f * x
2fx + 2fy = fx * f * y

Multiply fx * f
2fx + 2fy = f2x * y

Multiply f2x * y
2fx + 2fy = f2xy

Solving
2fx + 2fy = f2xy

Solving for variable 'f'.

Combine like terms: f2xy + -1f2xy = 0
2fx + 2fy + -1f2xy = 0

Factor out the Greatest Common Factor (GCF), 'f'.
f(2x + 2y + -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 '(2x + 2y + -1fxy)' equal to zero and attempt to solve:

Simplifying
2x + 2y + -1fxy = 0

Reorder the terms:
-1fxy + 2x + 2y = 0

Solving
-1fxy + 2x + 2y = 0

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

Add '-2x' to each side of the equation.
-1fxy + 2x + -2x + 2y = 0 + -2x

Combine like terms: 2x + -2x = 0
-1fxy + 0 + 2y = 0 + -2x
-1fxy + 2y = 0 + -2x
Remove the zero:
-1fxy + 2y = -2x

Add '-2y' to each side of the equation.
-1fxy + 2y + -2y = -2x + -2y

Combine like terms: 2y + -2y = 0
-1fxy + 0 = -2x + -2y
-1fxy = -2x + -2y

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

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

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

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