O(n) + O(n) = O(n)?

Question

According to Alex Martelli in O'Reilly's Python in a Nutshell, the complexity class O(n) + O(n) = O(n). So I believe it. But am confused. He explains it by saying that, "the sum of two linear functions of N is also a linear function of N."

According to wikipedia, in functional analysis a linear function is a linear map, an example of which would be f(x+y) = f(x) + f(y). Found what seems to be a simpler definition here simply stating, "a linear function is a function who's graph is a straight line." And it includes a few more examples that are less esoteric-looking than the wikipedia article ones.

y = f(x) = a + bx

and:

y = 25 + 5x

let x = 1
then
y = 25 + 5(1) = 30

let x = 3
then
y = 25 + 5(3) = 40

Maybe it would be fair to expect that the sum of two linear equations could be represented on a graph as a straight line which shows something similar to an average between the straight lines that would represent each one.

So am I understanding correctly that even though in the following two functions, the "complex" function takes three times as long to process as the "simple" one, they each function in big-O notation as O(n), because the arc of the processing time would be represented by a straight, diagonal line on a plot/graph, and the time difference would be represented by the fact that on a graphic representation, the angle of the more complex function would be sharper?

from timer import Timer

def complex(it):
    result = []
    for i in it:
        result.append(i)
    result.reverse()
    return result

def simple(it):
    result = list(it)

longlist = list(range(0, 1000000))

with Timer() as t:
    reverse_(longlist)
print "=> elapsed time reverse: %s." % t.secs

with Timer() as t:
    straight(longlist)
print "=> elapsed time straight: %s" % t.secs

Answer 1

Correct, O(n) + O(n) = O(n).

More specifically, O(n) + O(n) = 2 * O(n), but since Big O only cares about functions as they tend toward infinity, anything linear is denoted as O(n).

Answer 2

The statement is correct, because the addition of two linear functions is also a linear function. Take for example these two:

y = 6*x + 10
y = 20*x + 2

Add them together and you get:

y = 26*x + 12

Which is also a linear function! This holds for any two linear functions.

y = A*x + B
y = C*x + D
-----------
y = (A+C)*x + (B+D)