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

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

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

Reorder the terms for easier multiplication:
1f + f(f(1)) = 2

Reorder the terms for easier multiplication:
1f + f(1f) = 2

Remove parenthesis around (1f)
1f + f * 1f = 2

Reorder the terms for easier multiplication:
1f + 1f * f = 2

Multiply f * f
1f + 1f2 = 2

Solving
1f + 1f2 = 2

Solving for variable 'f'.

Reorder the terms:
-2 + 1f + 1f2 = 2 + -2

Combine like terms: 2 + -2 = 0
-2 + 1f + 1f2 = 0

Factor a trinomial.
(-2 + -1f)(1 + -1f) = 0

Subproblem 1
Set the factor '(-2 + -1f)' equal to zero and attempt to solve:

Simplifying
-2 + -1f = 0

Solving
-2 + -1f = 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 + -1f = 0 + 2

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

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

Divide each side by '-1'.
f = -2

Simplifying
f = -2
Subproblem 2
Set the factor '(1 + -1f)' equal to zero and attempt to solve:

Simplifying
1 + -1f = 0

Solving
1 + -1f = 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 + -1f = 0 + -1

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

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

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

Simplifying
f = 1
Solution
f = {-2, 1}