Enael wrote:

g(a+b,a+b) = g(a,a)+g(b,b) = f(a) + f(a) + f(b) + f(b)= 2f(a) + 2f(b) = RHS

g(a+b,a+b) = f(a+b)+f(a+b) = 2f(a+b) = LHS

We divide by 2 both RHS and LHS and get

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

I have an issue here: how do I know what to plug in from the answer choices? Since those are expressed in terms of X.

My approach was to take both a and b as the answer choice, but I don´t know if that is correct.

I am not sure what you mean by this last line but I can help you with the various variables.

The options (x+3), x^2 etc are the values of the function f(x)

Option (A) tells you that f(x) = x + 3

So if you want to find f(a) or f(b) or f(a+b), it is quite simple.

If f(x) = x+3, f(a) = a+3

If f(x) = x+3, f(a+b) = a+b+3

etc

Wherever you have x in the expression you put a or a+b or b as the case may be.

In this question, since options give the function f(x), you convert the entire g(x) into f(x).

You get that you need to find the function f(x) such that f(a+b) = f(a) + f(b)

The sum of individual functions of a and b and should be equal to the function of (a+b). We should look for an option where x is in the numerator and there is no addition/subtraction. So the first option I will try is (E)

If f(x) = x/4, f(a) = a/4, f(b) = b/4, f(a+b) = (a+b)/4

f(a) + f(b) = a/4 + b/4 = (a+b)/4

Hence, for option (E), f(a+b) = f(a) + f(b)

Answer (E)

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