What operator preserves the addition?

Question

 

Per example, these are operators that do not preserve the addition:

 

Ln (a+b) is not = ln a + ln b

like 

(a+b)^3 is not = a^3 + b^3

Maybe it has something with linear algebra since this question was asked by the teacher after determinants.

Answer 1

Multiplication by a constant preserves addition.  Example:

 

f(x) = mx

 

f(x₁ + x₂) = m(x₁ + x₂) = mx₁ + mx₂ = f(x₁) + f(x₂)