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

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For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] New post 12 Dec 2011, 05:52
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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

Can someone please tell me how to solve this? As OA is not provided I started by substituing the value of f(x) in the answer choice but got stuck. Can someone please help?
[Reveal] Spoiler: OA

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Re: function F(x) [#permalink] New post 12 Dec 2011, 06:47
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Question asks you to check which of the provided options satisfy the equality "f(a+b) = f(a) + f(b)"
Answer is E,
if you apply f(x) = -3x to f(a+b) then
f(a+b) = -3(a+b) = -3a -3b
f(a) = -3a
f(b) = -3b
f(a) + f(b) = -3a -3b = f(a+b)
Hope I am clear.
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Re: function F(x) [#permalink] New post 22 Dec 2011, 09:08
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f(a+b) has to be equal to f(a) + f(b)

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

A) f(x) = x^2
f(a) = a^2; f(b)=b^2
f(a+b) = (a+b)^2 = a^2+b^2 +2ab
f(a) +f(b) = a^2 + b^2 <> f(a+b)

so on and so forth...each option will lead to the same result except for E

Moreover, just by observing it can be found out the square roots, sqaures and x in the denominator will not be correct answers hence just try with E and you can find out..
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Re: function F(x) [#permalink] New post 15 Jan 2012, 15:06
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Responding to a 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) = \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.

Similar questions to practice:
function-85751.html
functions-problem-need-help-93184.html
let-the-function-g-a-b-f-a-f-b-143311.html

Hope it helps.
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Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] New post 16 Jan 2012, 01:00
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enigma123 wrote:
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

Can someone please tell me how to solve this? As OA is not provided I started by substituing the value of f(x) in the answer choice but got stuck. Can someone please help?


You can save time by using an intuitive method. Look for the expression that satisfies the distributive property i.e. x * (y + z) = (x * y) + (x * z)

When you put (a+b), it should give you individual functions in a and b which means that you will get two separate, comparable terms in a and b.
Squares, roots, addition and division by the variable does not satisfy the distributive property.
Multiplication does. So check for option (E) first.

One rule of thumb - in such questions, try the options which have multiplication/addition first. These two operators have various properties which make such relations possible.
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Re: Function Problem [#permalink] New post 07 Sep 2012, 22:17
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BigUp wrote:
Hello GmatClub,

Could someone please help me to understand this gmatprep question? Thanks!

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


First, a remark: a and b are considered positive because the function in C, the square root is not defined for negative numbers and the function in D is not defined for x=0 (x being in the denominator). The other functions are defined for any real number.

For each function, we translate the given equality and check whether is holds for any positive a and b. If the equality holds for any a and b, we should get an identity, which means the same expression on both sides of the equal sign.
For f(a+b) we take the expression of any of the given functions and replace x by (a+b).

(A) (a+b)^2=a^2+b^2 or a^2+2ab+b^2=a^2+b^2. Necessarily ab=0, which cannot hold, a an b being positive. NO
(B) a+b+1=a+1+b+1 gives 1=2, impossible. NO
(C) \sqrt{a+b}=\sqrt{a}+\sqrt{b} NO (check for example a=b=1)
(D) \frac{2}{a+b}=\frac{2}{b}+\frac{2}{b} NO (again, check for a=b=1)
(E) -3(a+b)=-3a+(-3b) or -3a-3b=-3a-3b YES!!!

Answer E.
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For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] New post 20 Dec 2012, 22:06
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Best to answer this with tiny-winie numbers such as a=1, b=1 and a+b=1...

A. (1) + (1) = 4 OUT!
B. (1+1) + (1+1) = 3 OUT!
C. 1 + 1 = \sqrt{2} OUT!
D. 2/1 + 2/1 = 2/2 OUT!
E. -3(1) -3(1) = -3(2) BINGO!

Answer: E
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Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] New post 08 Jul 2013, 00:10
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For which of the following functions is f(a+b) = fa + fb [#permalink] New post 06 Dec 2013, 08:30
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
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Re: For which of the following functions is f(a+b) = fa + fb [#permalink] New post 06 Dec 2013, 08:33
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tusharGupta1 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) = \sqrt{x}

(D) f(x) = \frac{2}{x}

(E) f(x) = -3x


Merging similar questions. Please refer to the solutions above.
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: For which of the following functions is f(a+b) = fa + fb   [#permalink] 06 Dec 2013, 08:33
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