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functions

by anuptvm » Thu Dec 23, 2010 8:23 pm
For which of the following functions is f(a+b) =f(a) +f(b) for all positive numbers a and b?

1. f(x) = x^2

2. f(x) = x+1

3. f(x) = SQRT(x)

4. f(x) =2/x

5. f(x) = =3x

I just couldn't understand what the problem was asking.
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by anshumishra » Thu Dec 23, 2010 8:42 pm
anuptvm wrote:For which of the following functions is f(a+b) =f(a) +f(b) for all positive numbers a and b?

1. f(x) = x^2

2. f(x) = x+1

3. f(x) = SQRT(x)

4. f(x) =2/x

5. f(x) = =3x

I just couldn't understand what the problem was asking.
Answer is 5.

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

So, f(a+b) = 3(a+b) = 3a + 3b = f(a) + f(b)

For the other functions, you can't prove that (I am proving just for one to get you the feeling) :
f(x) = x^2
f(a+b) = (a+b)^2 = a^2+2ab + b^2
f(a) = a^2
f(b) = b^2

Clearly, f(a+b) != f(a)+f(b)
Thanks
Anshu

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by Rahul@gurome » Thu Dec 23, 2010 8:46 pm
1 : f(a)=a^2 , f(b)=b^2, f(a+b)=(a+b)^2 which is not equal to (a^2) + (b^2) = f(a) + f(b)

2 : f(a)=a+1 , f(b)=b+1 , f(a+b)=a+b+1 which is not equal to (a+1)+(b+1) = a+b+2 = f(a) + f(b)

3 : f(a)= sqrt(a) , f(b)=sqrt(b) , f(a+b) = sqrt(a+b ) which is not equal to sqrt(a)+sqrt(b)= f(a) + f(b)

4 : f(a)=2/a , f(b) = 2/b , f(a+b)=2/(a+b) which is not equal to 2/a + 2/b = 2(a+b)/ab = f(a) + f(b)

5 : f(a)= 3a , f(b)= 3b , f(a+b)= 3(a+b)= 3a + 3b = f(a) + f(b)

The correct answer is 5.
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