We will have to solve for each option in this problem.
We can assume numbers or can just go ahead with a and b.
As Brent has already taken the assumption method, I would try to explain it by a and b

Option A:
f(a) = a^2, f(b) = b^2, so f(a)+f(b) = a^2+ b^2
f(a+b) = (a+b)^2 = a^2 + b^2 +2ab
This is not equal to f(a) + f(b)
So, we can rule out option A

Option B:
f(a)= a+1, f(b) = b+1, so f(a)+f(b) = a+b+2
f(a+b) = a+b+1
This is not equal to f(a) + f(b)
So, we can rule out option B

Option C:
f(a) = (a)^1/2, f(b) = (b)^1/2. So f(a)+f(b) = (a)^1/2+ (b)^1/2
f(a+b) = (a+b)^1/2
This is not equal to f(a) + f(b)
So, we can rule out option C

Option D:
f(a) = 2/a, f(b) = 2/b. So f(a)+f(b) = 2/a + 2/b = 2(a+b)/ab
f(a+b) = 2/(a+)
This is not equal to f(a) + f(b)
So, we can rule out option D

Hence, we are left with option E as the answer.
Still for confirmation, let us check it also

Option E:
f(a) = -3a, f(b) = -3b. So f(a)+f(b) = -3a + (-3b) = -3(a+b)
f(a+b) = -3(a+b)
This is equal to f(a) + f(b)
Hence E is the correct option.

Brent's solution is easy .

Initially I was surprised & think that something is missing in question

I got the question. Thanks Brent.