2f(x+1)=f(x)+f(2x)

Solution for 2f(x+1)=f(x)+f(2x) equation:

Simplifying
2f(x + 1) = f(x) + f(2x)

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

Multiply f * x
2f + 2fx = fx + f(2x)

Remove parenthesis around (2x)
2f + 2fx = fx + f * 2x

Reorder the terms for easier multiplication:
2f + 2fx = fx + 2f * x

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

Combine like terms: fx + 2fx = 3fx
2f + 2fx = 3fx

Solving
2f + 2fx = 3fx

Solving for variable 'f'.

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

Add '-3fx' to each side of the equation.
2f + 2fx + -3fx = 3fx + -3fx

Combine like terms: 2fx + -3fx = -1fx
2f + -1fx = 3fx + -3fx

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

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

Simplifying
2 + -1x = 0

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

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

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

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

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

Simplifying
0 = -2 + x

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
f = {0}