"+ to +" Maps: f(x+y) = f(x)+f(y) (Additive/Superposition)

Functions are additive

Some functions (though not all!) are additive. This means that if you add two numbers and then plug them into the function ( $f(x + y)$ ), you'll get the same answer as if you plugged two separate numbers into the function and then added the result ( $f(x) + f(y)$ ). To test if a function is additive, simply check that $f(x + y) = f(x) + f(y)$.

$f(4) = 11$ and $f(6) = 15$, so $f(4) + f(6) = 26$.

$4 + 6 = 10$, and $f(10) = 23$.

This function, a simple linear equation with a non-zero y-intercept, is not additive. Think about it some more: are there any types of functions that you think will always be additive?

See the explanation of Cauchy's functional equation for more information.