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

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

Simplifying
2f(x) * f(y) = f(x + y) + f(x + -1y)

Multiply f * x
2fx * f * y = f(x + y) + f(x + -1y)

Multiply fx * f
2f2x * y = f(x + y) + f(x + -1y)

Multiply f2x * y
2f2xy = f(x + y) + f(x + -1y)
2f2xy = (x * f + y * f) + f(x + -1y)
2f2xy = (fx + fy) + f(x + -1y)
2f2xy = fx + fy + (x * f + -1y * f)
2f2xy = fx + fy + (fx + -1fy)

Reorder the terms:
2f2xy = fx + fx + fy + -1fy

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

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

Solving
2f2xy = 2fx

Solving for variable 'f'.

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

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

Factor out the Greatest Common Factor (GCF), '2fx'.
2fx(-1 + fy) = 0

Ignore the factor 2.
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 '(-1 + fy)' equal to zero and attempt to solve:

Simplifying
-1 + fy = 0

Solving
-1 + fy = 0

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

Add '1' to each side of the equation.
-1 + 1 + fy = 0 + 1

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

Combine like terms: 0 + 1 = 1
fy = 1

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

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