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

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

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

Reorder the terms:
fx + fx + fy + -1fy = 2f(x) * f(y)

Combine like terms: fx + fx = 2fx
2fx + fy + -1fy = 2f(x) * f(y)

Combine like terms: fy + -1fy = 0
2fx + 0 = 2f(x) * f(y)
2fx = 2f(x) * f(y)

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

Multiply fx * f
2fx = 2f2x * y

Multiply f2x * y
2fx = 2f2xy

Solving
2fx = 2f2xy

Solving for variable 'f'.

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

Factor out the Greatest Common Factor (GCF), '2fx'.
2fx(1 + -1fy) = 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 + -1fy)' equal to zero and attempt to solve:

Simplifying
1 + -1fy = 0

Solving
1 + -1fy = 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 + -1fy = 0 + -1

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

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

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

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