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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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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?

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Re: function F(x)  [#permalink]

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New post 15 Jan 2012, 15:06
85
88
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: function F(x)  [#permalink]

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New post 12 Dec 2011, 06:47
15
8
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]

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New post 22 Dec 2011, 09:08
2
1
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: For which of the following functions is f(a+b)=f(b)+f(a)  [#permalink]

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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]

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New post 07 Sep 2012, 22:17
1
2
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]

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New post 20 Dec 2012, 22:06
2
3
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]

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New post 08 Jul 2013, 00:10
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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

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

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

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New post 15 Jan 2016, 08:59
3
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/
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For which of the following functions is f(a+b)=f(b)+f(a)  [#permalink]

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New post 06 Apr 2016, 12:23
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
Screen Shot 2016-08-04 at 6.53.30 PM.png [ 112.94 KiB | Viewed 99254 times ]

Screen Shot 2016-08-04 at 7.22.53 PM.png
Screen Shot 2016-08-04 at 7.22.53 PM.png [ 792.38 KiB | Viewed 99402 times ]

IMG_1002.JPG
IMG_1002.JPG [ 324.37 KiB | Viewed 99255 times ]


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

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New post 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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New post 11 Oct 2017, 10:32
1
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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New post 08 Jan 2018, 10:32
1
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)
Therefore e) is the answer.
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Re: For which of the following functions is f(a+b)=f(b)+f(a)  [#permalink]

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New post 06 Feb 2018, 14:50
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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


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

The correct answer is E

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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New post 12 Feb 2018, 11:01
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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


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.

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