f(a+b) = f(a) + f(b)

Insert a and b in the answer options

f(x) = x^2

f(a+b) = (a+b)^2

f(a) = a^2
f(b) = b^2

(a+b)^2 is not equal to a^2 + b^2

Therefore we have to find the value for f(x) which hold true for f(a+b) = f(a) + f(b)

After pluggin in the a and b for x in all the options you will find that only Option E holds true.

f(x) = -3x

f(a+b)= -3(a+b) = -3a - 3b

f(a) = -3a
f(b) = -3b

-3a - 3b = -3a + (-3b) = -3a - 3b

Hence E is the answer