f(f(x))=f(x-3)

Solution for f(f(x))=f(x-3) equation:

Simplifying
f(f(x)) = f(x + -3)

Multiply f * x
f(fx) = f(x + -3)

Multiply f * fx
f2x = f(x + -3)

Reorder the terms:
f2x = f(-3 + x)
f2x = (-3 * f + x * f)
f2x = (-3f + fx)

Solving
f2x = -3f + fx

Solving for variable 'f'.

Reorder the terms:
3f + -1fx + f2x = -3f + fx + 3f + -1fx

Reorder the terms:
3f + -1fx + f2x = -3f + 3f + fx + -1fx

Combine like terms: -3f + 3f = 0
3f + -1fx + f2x = 0 + fx + -1fx
3f + -1fx + f2x = fx + -1fx

Combine like terms: fx + -1fx = 0
3f + -1fx + f2x = 0

Factor out the Greatest Common Factor (GCF), 'f'.
f(3 + -1x + fx) = 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 '(3 + -1x + fx)' equal to zero and attempt to solve:

Simplifying
3 + -1x + fx = 0

Reorder the terms:
3 + fx + -1x = 0

Solving
3 + fx + -1x = 0

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

Add '-3' to each side of the equation.
3 + fx + -3 + -1x = 0 + -3

Reorder the terms:
3 + -3 + fx + -1x = 0 + -3

Combine like terms: 3 + -3 = 0
0 + fx + -1x = 0 + -3
fx + -1x = 0 + -3

Combine like terms: 0 + -3 = -3
fx + -1x = -3

Add 'x' to each side of the equation.
fx + -1x + x = -3 + x

Combine like terms: -1x + x = 0
fx + 0 = -3 + x
fx = -3 + x

Divide each side by 'x'.
f = -3x-1 + 1

Simplifying
f = -3x-1 + 1

Reorder the terms:
f = 1 + -3x-1
Solution
f = {0, 1 + -3x-1}