For which of the following function is f(a+b)=f(a)+f(b) for all positive numbers a and b?

(A) f(x)=x^2

(B) f(x)=x+1

(C) f(x)=√x

(D) f(x)=2/x

(E) f(x)=-3x

Plug in a=2, b=3.

f(a+b) = f(2+3) = f(5)

Then the question becomes:

For which of the following functions does f(5) = f(2) + f(3)?

Answer choice A:
f(5) = 5² = 25
f(2) = 2² = 4
f(3) = 3² = 9
25 = 4+9 Doesn't work.

Answer choice B:
f(5) = 5+1 = 6
f(2) = 2+1 = 3
f(3) = 3+1 = 4
6 = 3+4 Doesn't work.

Answer choice C:
f(5) = √5
f(2) = √2
f(3) = √3
√5 = √2 + √3 Doesn't work.

Answer choice D:
f(5) = 2/5
f(2) = 2/2 = 1
f(3) = 2/3
2/5 = 1 + 2/3 Doesn't work.

The correct answer is E.

Answer choice E:
f(5) = -3*5 = -15
f(2) = -3*2 = -6
f(3) = -3*3 = -9
-15 = -6 + -9
Success!