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For which of the following functions is f(a+b)=f(b)+f(a) for

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For which of the following functions is f(a+b)=f(b)+f(a) for [#permalink] New post 30 Aug 2006, 10:33
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For which of the following functions is f(a+b)=f(b)+f(a) for all positive numbers a and b?

A. f(x)=x^2
B. f(x)=x+1
C. f(x)=√x
D. f(x)=2/x
E. f(x)=-3x

OPEN DISCUSSION OF THIS QUESTION IS HERE: for-which-of-the-following-functions-is-f-a-b-f-b-f-a-124491.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 28 Jul 2014, 12:57, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: For which of the following functions is f(a+b)=f(b)+f(a) for [#permalink] New post 30 Aug 2006, 10:47
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lan583 wrote:
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. of x
f(x)= 2/x
f(x)=-3x


Straight E.

1. (a+b)^2 a^2+b^2
2. a+b+1 (a+1) + (b+1)
3. √(a+b) √a + √b
4. 2/(a+b) 2/a + 2/b
5. -3(a+b) = -3a + (-3b)
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Re: For which of the following functions is f(a+b)=f(b)+f(a) for [#permalink] New post 28 Jul 2014, 12:48
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Re: For which of the following functions is f(a+b)=f(b)+f(a) for [#permalink] New post 28 Jul 2014, 12:59
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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) = \sqrt{x}
D. f(x)=\frac{2}{x}
E. f(x) = -3x

A. f(a+b)=(a+b)^2=a^2+2ab+b^2\neq{f(a)+f(b)}=a^2+b^2

B. f(a+b)=(a+b)+1\neq{f(a)+f(b)}=a+1+b+1

C. f(a+b)=\sqrt{a+b}\neq{f(a)+f(b)}=\sqrt{a}+\sqrt{b}.

D. f(a+b)=\frac{2}{a+b}\neq{f(a)+f(b)}=\frac{2}{a}+\frac{2}{b}.

E. f(a+b)=-3(a+b)=-3a-3b=f(a)+f(b)=-3a-3b. Correct.

Answer: E.

OR, as f(a+b)= f(a)+f(b) must be true for all positive numbers a and b, then you can randomly pick particular values of a and b and check for them:

For example: a=2 and b=3

A. f(a + b) = f(5) = 5^2 = 25\neq{f(a) + f(b) = f(2) + f(3) = 2^2 + 3^2 = 13}

B. f(a + b) = f(5) = 5 + 1 = 6\neq{f(a) + f(b) = f(2) + f(3) = (2 + 1) + (3 + 1) = 7}

C. f(a + b)=f(5)=\sqrt{5}\neq{f(a) + f(b)=f(2)+f(3)=\sqrt{2}+\sqrt{3}}

D. f(a + b)=f(5)=\frac{2}{5}\neq{f(a)+f(b)=f(2)+f(3)=\frac{2}{2}+\frac{2}{3}=\frac{5}{3}}

E. f(a + b)=f(5)=-3*(5) =-15=f(a)+f(b)=f(2)+f(3)=-3*(2)-3*(3)=-15. Correct.

It might happen that for some choices of a and b other options may be "correct" as well. If this happens just pick some other numbers and check again these "correct" options only.

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OPEN DISCUSSION OF THIS QUESTION IS HERE: for-which-of-the-following-functions-is-f-a-b-f-b-f-a-124491.html
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Re: For which of the following functions is f(a+b)=f(b)+f(a) for   [#permalink] 28 Jul 2014, 12:59
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