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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]

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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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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.

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]

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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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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 18112 [14], given: 236 Manager Joined: 12 Jul 2011 Posts: 139 Kudos [?]: 53 [10], given: 42 Concentration: Operations, Strategy GMAT 1: 680 Q46 V37 WE: Engineering (Telecommunications) Re: function F(x) [#permalink] ### Show Tags 12 Dec 2011, 06:47 10 This post received KUDOS 5 This post was BOOKMARKED 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. Kudos [?]: 53 [10], given: 42 Manager Joined: 09 Nov 2011 Posts: 127 Kudos [?]: 67 [2], given: 16 Re: function F(x) [#permalink] ### Show Tags 22 Dec 2011, 09:08 2 This post received KUDOS 1 This post was BOOKMARKED 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.. _________________ Time to play the game... Kudos [?]: 67 [2], given: 16 Senior Manager Joined: 13 Aug 2012 Posts: 457 Kudos [?]: 570 [2], given: 11 Concentration: Marketing, Finance GPA: 3.23 For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] ### Show Tags 20 Dec 2012, 22:06 2 This post received KUDOS 2 This post was BOOKMARKED 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 _________________ Impossible is nothing to God. Kudos [?]: 570 [2], given: 11 Math Expert Joined: 02 Sep 2009 Posts: 42585 Kudos [?]: 135506 [2], given: 12697 Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] ### Show Tags 08 Jul 2013, 00:10 2 This post received KUDOS Expert's post Bumping for review and further discussion*. Get a kudos point for an alternative solution! *New project from GMAT Club!!! Check HERE To find DS questions by Kudos, sort by Kudos here: gmat-data-sufficiency-ds-141/ To find PS questions by Kudos, sort by Kudos here: gmat-problem-solving-ps-140/ _________________ Kudos [?]: 135506 [2], given: 12697 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7792 Kudos [?]: 18112 [2], given: 236 Location: Pune, India Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] ### Show Tags 15 Jan 2016, 08:59 2 This post received KUDOS Expert's post pepo wrote: Bunuel wrote: 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. I always get confused with these kind of questions and I like the method to pick numbers to check whether answer choices are equal to the main statement. Here are a couple of posts on functions. They could help you. http://www.veritasprep.com/blog/2015/03 ... s-on-gmat/ http://www.veritasprep.com/blog/2015/03 ... questions/ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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07 Sep 2012, 22:17
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BigUp wrote:
Hello GmatClub,

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!!!

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Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]

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27 Feb 2015, 02:27
sagarag wrote:
For which of the following functions is F(a+b) = f(a) + f(b) for all positives 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

hi sagar..
you can eliminate the choices by looking at the choices..
the answer cannot be a variable added or subtracted with a constant.. since that value will get added/subtracted twice on right side... B is out
it cannot be a variable multiplied with another variable or with self.. A and C out..
the answer can be a variable multiplied or divided by a constant...
D is out as a constant is divided by variable.. E follows the above rule
ans E..
you can also find answer by testing values ..
take E for example..
since f(x) = -3x, f(a) = -3a f(b) = -3b.. f(a+b)=-3(a+b)=-3a+(-3b)=f(a)+f(b)... E is the ans
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]

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15 Jan 2016, 06:37
Bunuel wrote:
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.

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.

I always get confused with these kind of questions and I like the method to pick numbers to check whether answer choices are equal to the main statement.

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Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]

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24 Mar 2016, 03:54
For which of the following functions is f(a+b)=f(a)f(b) for all real numbers a and b?
A) f(x) = sin(x )
B) f( x) =log3x
C) f( x) =e^x
D) f(x)= x

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Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]

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24 Mar 2016, 04:06
For which of the following functions is f(a+b)=f(a)f(b) for all real numbers a and b?
A) f(x) = sin(x )
B) f( x) =log3x
C) f( x) =e^x
D) f(x)= x

Hi,

firstly the trignometric expressions are not tested in GMAT
secondly, pl post new topics separately by following guidelines
if f( x) =e^x
f( a) =e^a
f( b) =e^b..
so f(a)f(b)= e^a*e^b= e^(a+b)= f(a+b)
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]

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24 Mar 2016, 04:13
For which of the following functions is f(a+b)=f(a)f(b) for all real numbers a and b?
A) f(x) = sin(x )
B) f( x) =log3x
C) f( x) =e^x
D) f(x)= x

This is NOT a GMAT question. So, you can ignore it.
_________________

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For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]

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06 Apr 2016, 12:23
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Attached is a visual that should help. Alternate method: plug in a=1 and b=1 to make the math easier.
Attachments

Screen Shot 2016-08-04 at 6.53.30 PM.png [ 112.94 KiB | Viewed 52931 times ]

Screen Shot 2016-08-04 at 7.22.53 PM.png [ 792.38 KiB | Viewed 52984 times ]

IMG_1002.JPG [ 324.37 KiB | Viewed 52912 times ]

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For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]

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11 Oct 2017, 10:20
VeritasPrepKarishma wrote:
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.

a golden rule of thumb: trying multiplication/ addition first

I have applied it to couple of cases, and the rule duly worked, saving time

thanks, mam

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Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]

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11 Oct 2017, 10:32
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?

hi

A: (a + b) ^ 2 is not equal to a^ + b ^2, discard
B: a + b + 1 is not equl to b + 1 + a + 1, discard
C: square rootover (a + b) is not equal to rootover a + rootover b , discard
D: 2/ a+b is not equal to 2/a + 2/b
E: -3 (a + b) is equal to -3a -3b, so this is the correct choice

thanks

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Re: For which of the following functions is f(a+b)=f(b)+f(a)   [#permalink] 11 Oct 2017, 10:32
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