f(f(2))=f(2x)

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

Simplifying
f(f(2)) = f(2x)

Reorder the terms for easier multiplication:
f(2f) = f(2x)

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

Reorder the terms for easier multiplication:
2f * f = f(2x)

Multiply f * f
2f2 = f(2x)

Remove parenthesis around (2x)
2f2 = f * 2x

Reorder the terms for easier multiplication:
2f2 = 2f * x

Multiply f * x
2f2 = 2fx

Solving
2f2 = 2fx

Solving for variable 'f'.

Reorder the terms:
-2fx + 2f2 = 2fx + -2fx

Combine like terms: 2fx + -2fx = 0
-2fx + 2f2 = 0

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

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

Simplifying
-1x + f = 0

Reorder the terms:
f + -1x = 0

Solving
f + -1x = 0

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

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

Combine like terms: -1x + x = 0
f + 0 = 0 + x
f = 0 + x
Remove the zero:
f = x

Simplifying
f = x
Solution
f = {0, x}