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functions

by rerika01 » Tue Jul 05, 2011 7:03 pm
For which of the following functions is f(a+b)=f(a)+f(b) for all positive numbers "a" and "b"?

a) f(x)= x^2
b) f(x)=x+1
c) f(x)=x^1/2
d) f(x)=2/x
e) f(x)=-3x

This is a question from one of the practice tests. Can anyone share quick strategy to approach a problem like this?
The answer to this question is e)

Thanks!
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by abhisays » Tue Jul 05, 2011 7:14 pm
Let us take one example say a = 2 and b = 3, now we will work on option 1 which becomes f(2+3) = f(2)+f(3) means sqaure of 5 on L.H.S and square of 2 + square of 3 on R.H.S..

Now L.H.S we have 25 and on R.H.S we have 9 so option A is eliminated.

Similarly we can eliminate option B,C,D

Now work on option E.

L.H.S = f(5) = -15

R.H.S = f(2) + f (3) = -6 + (-9) = -15

Hence L.H.S = R.H.S

So the correct answer will be (e)
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by GMATGuruNY » Tue Jul 05, 2011 7:33 pm
rerika01 wrote:For which of the following functions is f(a+b)=f(a)+f(b) for all positive numbers "a" and "b"?

a) f(x)= x^2
b) f(x)=x+1
c) f(x)=x^1/2
d) f(x)=2/x
e) f(x)=-3x

This is a question from one of the practice tests. Can anyone share quick strategy to approach a problem like this?
The answer to this question is e)

Thanks!
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by Anurag@Gurome » Tue Jul 05, 2011 9:09 pm
rerika01 wrote:For which of the following functions is f(a+b)=f(a)+f(b) for all positive numbers "a" and "b"?

a) f(x)= x^2
b) f(x)=x+1
c) f(x)=x^1/2
d) f(x)=2/x
e) f(x)=-3x
Easiest method to solve this is to use the concept of linearity. If f(a + b) = f(a) + f(b), then the function f(.) must be linear. Only option B and E are linear functions. Now check them individually.

For option B, f(a) = (a + 1) and f(b) = (b + 1). Hence, f(a) + f(b) = (a + b + 2). But f(a + b) = (a + b + 1) ≠ f(a) + f(b)

Then option E is our only choice.

The correct answer is E.
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