Functions

[DCJ]

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

What's the best way to solve this without trying each answer choice? Had I looked at choice E first, I would have recognized the use of distributive property and that E was the right answer but instead I started by trying all the answer choices in order.

[Jim@StratusPrep]

There really is no other way than looking at each answer. The key is to have a grasp of what principles each answer induces so you don't have to do the math for each.

a) eliminates middle term from FOIL
b) add 1 twice
c) same as a
d) common denominator will change numerator
e) your answer

[Anurag@Gurome]

For which of the following functions is f(a+b)=f(a)+f(b) for all positive numbers a and b?
...
What's the best way to solve this without trying each answer choice?

The given relation f(a + b) = f(a) + f(b) is the additive property which holds for most of the linear relations. If you know this you'll check for the linear equations first.

Now, we have only two linear relations f(x) = (x + 1) and f(x) = -3x.
Even without checking each of them, we can see that the former adds an extra 1.

Now, check for f(x) = -3x ---> f(a + b) = -3(a + b) = -3a -3b = f(a) + f(b)