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

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

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

Multiply f * x
fx * f * y = f(x + y) + f(x + -1y)

Multiply fx * f
f2x * y = f(x + y) + f(x + -1y)

Multiply f2x * y
f2xy = f(x + y) + f(x + -1y)
f2xy = (x * f + y * f) + f(x + -1y)
f2xy = (fx + fy) + f(x + -1y)
f2xy = fx + fy + (x * f + -1y * f)
f2xy = fx + fy + (fx + -1fy)

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

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

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

Solving
f2xy = 2fx

Solving for variable 'f'.

Reorder the terms:
-2fx + f2xy = 2fx + -2fx

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

Factor out the Greatest Common Factor (GCF), 'fx'.
fx(-2 + fy) = 0

Subproblem 1
Set the factor 'fx' equal to zero and attempt to solve:

Simplifying
fx = 0

Solving
fx = 0

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

Simplifying
fx = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(-2 + fy)' equal to zero and attempt to solve:

Simplifying
-2 + fy = 0

Solving
-2 + fy = 0

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

Add '2' to each side of the equation.
-2 + 2 + fy = 0 + 2

Combine like terms: -2 + 2 = 0
0 + fy = 0 + 2
fy = 0 + 2

Combine like terms: 0 + 2 = 2
fy = 2

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

Simplifying
f = 2y-1
Solution
f = {2y-1}