Basically eliminate functions. This can be done really quickly, it really takes more to try to write them on paper then solving it:

a. f(a + b) = (a + b)^2 = a^2 + b^2 + 2ab = f(a) + f(b) + 2ab - out

b. f(a + b) = a + b + 1 = a + 1 + b = f(a) + f(b) - 1 - out

c. f(a + b) = sqrt(a + b), which is almost always different from sqrt(a) + sqrt(b) - pick 9 and 16 to test that

d. f(a + b) = 2/(a + b), while f(a) + f(b) = 2/a + 2/b = 2(a + b)/ab - almost always different from 2/(a + b)

e. is the only one left:
f(a + b) = -3(a + b) = -3a - 3b = f(a) + f(b)

Great explanation! Thanks!