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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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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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Joined: 09 Nov 2011
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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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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 Director Joined: 22 Mar 2011 Posts: 608 WE: Science (Education) Re: Function Problem [#permalink] ### Show Tags 07 Sep 2012, 22:17 2 This post received KUDOS 1 This post was BOOKMARKED 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: 456 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. Math Expert Joined: 02 Sep 2009 Posts: 43804 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/ _________________ Math Expert Joined: 02 Aug 2009 Posts: 5651 Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] ### Show Tags 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 _________________ 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 BANGALORE/- Manager Joined: 21 Jan 2014 Posts: 99 GMAT 1: 500 Q32 V28 GPA: 4 Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink] ### Show Tags 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: 7938 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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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.
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Screen Shot 2016-08-04 at 6.53.30 PM.png [ 112.94 KiB | Viewed 60986 times ]

Screen Shot 2016-08-04 at 7.22.53 PM.png [ 792.38 KiB | Viewed 61049 times ]

IMG_1002.JPG [ 324.37 KiB | Viewed 60965 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]

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08 Jan 2018, 10:32
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

I substituted x in a and b -> f(a+b)= f(a) + f(b) --> f(x+x)= f(x) + f(x) --> f(2x)= 2 f(x)
so I have to find the option that satisfies this equation.
a) f(2x)= 4x^2 vs 2f(x)=2x^2 ( Not equal)
b) f(2x)=2x+1 vs 2f(x)= 2x+2 ( Not equal)
c) f(2x)= sqrt 2*sqrt x vs 2f(x)= 2 * sqrt x (Not equal)
d) f(2x)=1/x vs 2f(x)= 4/x (Not equal)
e) f(2x)=-6x vs 2f(x)= -6x (equal)
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Re: For which of the following functions is f(a+b)=f(b)+f(a) [#permalink]

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06 Feb 2018, 14:50
Expert's post
Top Contributor
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

One approach is to plug in numbers. Let's let a = 1 and b = 1

So, the question becomes, "Which of the following functions are such that f(1+1) = f(1) + f(1)?"
In other words, for which function does f(2) = f(1) + f(1)?

A) If f(x)=x², does f(2) = f(1) + f(1)?
Plug in to get: 2² = 1² + 1²? (No, doesn't work)
So, it is not the case that f(2) = f(1) + f(1), when f(x)=x²

B) If f(x)=x+1, does f(2) = f(1) + f(1)?
Plug in to get: 2+1 = 1+1 + 1+1? (No, doesn't work)
So, it is not the case that f(2) = f(1) + f(1)
.
.
.
A, B, C and D do not work.
So, at this point, we can conclude that E must be the correct answer.
Let's check E anyway (for "fun")

E) If f(x)=-3x, does f(2) = f(1) + f(1)?
Plugging in 2 and 1 we get: (-3)(2) = (-3)(1) + (-3)(1)
Yes, it works

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

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12 Feb 2018, 11:01
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

We need to determine when f(a + b) = f(a) + f(b). Before we evaluate each answer choice it may be easier to use numerical values for a and b. If we let a = 1 and b = 2, our new function looks like:

f(1 + 2) = f(1) + f(2)

f(3) = f(1) + f(2)

So we must determine when the output of f(3) equals the sum of the outputs of f(1) and f(2).

Let’s now evaluate each answer choice.

A) f(x) = x^2

f(3) = 3^2 = 9

f(1) = 1^2 = 1

f(2) = 2^2 = 4

Since 9 does not equal 1 + 4, choice A is not correct.

B) f(x) = x + 1

f(3) = 3 + 1 = 4

f(1) = 1 + 1 = 2

f(2) = 2 + 1 = 3

Since 4 does not equal 2 + 3, choice B is not correct.

C) f(x) = √x

f(3) = √3

f(1) = √1 = 1

f(2) = √2

Since √3 does not equal 1 + √2, choice C is not correct.

D) f(x) = 2/x

f(3) = 3/2

f(1) = 2/1 = 2

f(2) = 2/2 = 1

Since 3/2 does not equal 2 + 1, choice D is not correct.

Since we have eliminated all the other answer choices, we know the answer is E. However, let’s show as an exercise that answer choice E satisfies the given property for our choice of numbers:

E) f(x) = -3x

f(3) = -3(3) = -9

f(1) = -3(1) = -3

f(2) = -3(2) = -6

Since -9 equals -3 + (-6), choice E is correct.

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Re: For which of the following functions is f(a+b)=f(b)+f(a)   [#permalink] 12 Feb 2018, 11:01
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