f(3x+2y)=f(x)f(y)

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Solution for f(3x+2y)=f(x)f(y) equation: 


Simplifying
f(3x + 2y) = f(x) * f(y)
(3x * f + 2y * f) = f(x) * f(y)
(3fx + 2fy) = f(x) * f(y)

Multiply f * x
3fx + 2fy = fx * f * y

Multiply fx * f
3fx + 2fy = f2x * y

Multiply f2x * y
3fx + 2fy = f2xy

Solving
3fx + 2fy = f2xy

Solving for variable 'f'.

Combine like terms: f2xy + -1f2xy = 0
3fx + 2fy + -1f2xy = 0

Factor out the Greatest Common Factor (GCF), 'f'.
f(3x + 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 '(3x + 2y + -1fxy)' equal to zero and attempt to solve:

Simplifying
3x + 2y + -1fxy = 0

Reorder the terms:
-1fxy + 3x + 2y = 0

Solving
-1fxy + 3x + 2y = 0

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

Add '-3x' to each side of the equation.
-1fxy + 3x + -3x + 2y = 0 + -3x

Combine like terms: 3x + -3x = 0
-1fxy + 0 + 2y = 0 + -3x
-1fxy + 2y = 0 + -3x
Remove the zero:
-1fxy + 2y = -3x

Add '-2y' to each side of the equation.
-1fxy + 2y + -2y = -3x + -2y

Combine like terms: 2y + -2y = 0
-1fxy + 0 = -3x + -2y
-1fxy = -3x + -2y

Divide each side by '-1xy'.
f = 3y-1 + 2x-1

Simplifying
f = 3y-1 + 2x-1

Reorder the terms:
f = 2x-1 + 3y-1
Solution
f = {0, 2x-1 + 3y-1}

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