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

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Director
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For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]  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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Manager
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Kudos [?]: 31 [4] , given: 42

Re: function F(x) [#permalink]  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)"
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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Kudos [?]: 47 [2] , given: 16

Re: function F(x) [#permalink]  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]  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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Kudos [?]: 9657 [7] , given: 198

Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]  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 Director Joined: 22 Mar 2011 Posts: 612 WE: Science (Education) Followers: 86 Kudos [?]: 720 [1] , given: 43 Re: Function Problem [#permalink] 07 Sep 2012, 22:17 1 This post received KUDOS 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. _________________ PhD in Applied Mathematics Love GMAT Quant questions and running. Senior Manager Joined: 13 Aug 2012 Posts: 464 Concentration: Marketing, Finance GMAT 1: Q V0 GPA: 3.23 Followers: 20 Kudos [?]: 321 [1] , given: 11 For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] 20 Dec 2012, 22:06 1 This post received KUDOS 1 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. Math Expert Joined: 02 Sep 2009 Posts: 31304 Followers: 5365 Kudos [?]: 62546 [0], given: 9457 Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] 08 Jul 2013, 00:10 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/ _________________ GMAT Club Legend Joined: 09 Sep 2013 Posts: 8237 Followers: 419 Kudos [?]: 111 [0], given: 0 Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] 09 Dec 2014, 10:41 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Intern Joined: 13 May 2014 Posts: 30 GMAT Date: 11-01-2014 Followers: 0 Kudos [?]: 9 [0], given: 43 Functions [#permalink] 26 Feb 2015, 11:42 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 Math Expert Joined: 02 Sep 2009 Posts: 31304 Followers: 5365 Kudos [?]: 62546 [0], given: 9457 Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] 26 Feb 2015, 11:44 Expert's post 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 Merging similar questions. Please refer to the solutions above. P.S. Please name topics properly. Check ruke 3 here: rules-for-posting-please-read-this-before-posting-133935.html _________________ Verbal Forum Moderator Joined: 02 Aug 2009 Posts: 2068 Followers: 82 Kudos [?]: 965 [0], given: 37 Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] 27 Feb 2015, 02:27 Expert's post 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 _________________ BACK IN CLUB AFTER A GAP OF 6 YEARS http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 Errors in CR ---- http://gmatclub.com/forum/gmat-institutes-please-review-the-following-qs-211573.html Manager Joined: 21 Jan 2014 Posts: 70 Followers: 1 Kudos [?]: 5 [0], given: 31 Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] 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. 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. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6238 Location: Pune, India Followers: 1683 Kudos [?]: 9657 [1] , given: 198 Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] 15 Jan 2016, 08:59 1 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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Re: For which of the following functions is f(a+b)=f(b)+f(a)   [#permalink] 15 Jan 2016, 08:59
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# For which of the following functions is f(a+b)=f(b)+f(a)

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