F(a+b) = F(a)+F(b)

[AshvinV429]

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


f(x) = x^ 2
f(x) = x + 1
f(x) = Sqrt (x)
f(x) = 2/x
f(x) = –3x

I was able to find the solution as (e) but had to go through each answer and substitute numbers for a & b. is there an easier solution?

[Sage Pearce-Higgins]

Using numbers (testing cases) is a good strategy for this kind of formula problem; it's worth being fluent at that strategy so that you can use is quickly and efficiently. However, the algebraic route isn't bad here either. For example, answer A would be: does (a+b)^2 = a^2 + b^2 ? If you know your exponent rules, then you'll know that this equation doesn't have to be true. You might even see what the question is asking in words: "Is it the same if you do the function once on the sum of a and b and if you do the function individually on each of a and b and then add them together?" This might help you eliminate answer b, as you effectively "do the function twice" and therefore add on 2. In reality, an efficient strategy for a problem like this would probably combine all three approaches, eliminating some answers quite quickly, and then testing cases for the last couple of answers.