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

Question

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 11.JPG

Bunuel · Accepted Answer

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:
https://gmatclub.com/forum/for-which-of- ... 24491.html
https://gmatclub.com/forum/for-which-of- ... 85751.html
https://gmatclub.com/forum/let-the-funct ... 43311.html

Hope it helps.

Bunuel · Answer

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:
https://gmatclub.com/forum/for-which-of- ... 24491.html
https://gmatclub.com/forum/for-which-of- ... 85751.html
https://gmatclub.com/forum/let-the-funct ... 43311.html

Hope it helps.

KarishmaB · Answer

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.

arjunbt · Answer

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.

BrentGMATPrepNow · Answer

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:  2 ² =  1 ² +  1 ²? (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

tonebeeze · Answer

Thanks for really fleshing out the algebra on this problem Bunuel. The problems seem fairly easy once you understand how to work functions properly.

iamgame · Answer

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..

EvaJager · Answer

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.

mbaiseasy · Answer

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

pepo · Answer

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.

KarishmaB · Answer

Here are a couple of posts on functions. They could help you. https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2015/03 ... s-on-gmat/ https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2015/03 ... questions/

mcelroytutoring · Answer

Attached is a visual that should help. Alternate method: plug in a=1 and b=1 to make the math easier.

testcracker · Answer

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

kavita9 · Answer

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.

JeffTargetTestPrep · Answer

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

CrackverbalGMAT · Answer

Solution:

Use the options to answer the question

A. f(a+b)=(a+b)^2=a^2+2ab+b^2 and this is NOT equal to f(a)+f(b) which is a^2+b^2 (Eliminate)

B. f(a+b)=(a+b)+1 and this is NOT equal to f(a)+f(b) which =a+1+b+1=a+b+2 (Eliminate)

C. f(a+b)=√(a+b) and this is NOT equal to f(a)+f(b) which is = √a+√b (Eliminate)

D. f(a+b)=2/a+b and this is NOT equal to f(a)+f(b) which is =2/a+2/b (Eliminate)

E. f(a+b)=−3(a+b)=−3a−3b and this is EQUAL to f(a)+f(b)=−3a−3b.  (option e )

Devmitra Sen
GMAT SME

avigutman · Answer

Video solution from Quant Reasoning starts at 8:40 Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=CDT-CCLAaHY

BrushMyQuant · Answer

Given that f(a + b) = f(a) + f(b) and we need to find which of the following can be the value of f(x) which satisfies this.

Let's solve the problem using two methods

Method 1: Logic (Eliminate Option Choices)

f(a+b) = f(a) + f(b)

Now, this can be true only when 
  1.  We don't have any constant term added or subtracted from any term of x. As if we have one then on Left Hand Side(LHS) that constant term will be added or subtracted only once, but on Right Hand Side(RHS) it will be added or subtracted twice.
 2.  We don't have x in the denominator (in general) as we wont be able to match the LHS and RHS then.
 3.  We don't have any power of x ≠ 1 in the numerator. As otherwise (in general) we wont be able to match the LHS and RHS.
 4.  We have a term of x in the numerator with power of 1 with any positive or negative constant multiplied with it. Ex 2x, -3x, etc

Using above logic we can eliminate the answer choices

(A)  f(x)=x^2  
=>   Eliminate   : Doesn't Satisfy Point 3 above. Power of x is 2.

(B)  f(x)= x+1  
=>   Eliminate   : Doesn't Satisfy Point 1 above. It has a constant added (+1)

(C)  \sqrt{x}  
=>   Eliminate   : Doesn't Satisfy Point 3 above. Power of x is  \frac{1}{2}

(D)  f(x)=\frac{2}{x}  
=>   Eliminate   : Doesn't Satisfy Point 2 above. x is in denominator

(E)  -3x  
=>   POSSIBLE  : Satisfies all the conditions above.

So,  Answer will be E .

Method 2: Algebra (taking all option choices)

(A)  f(x)=x^2

To find f(a+b) we need to compare what is inside the bracket in f(a+b) and f(x)
=> We need to substitute x with a+b in  f(x)=x^2  to get the value of f(a+b)

=> f(a+b) =  (a+b)^2  =  a^2 + 2ab + b^2 
f(a) =  a^2  and f(b) =  b^2 
=> f(a) + f(b) =  a^2  +  b^2  =  a^2 + b^2  ≠  a^2 + 2ab + b^2 
=> f(a+b) ≠ f(a) + f(b) =>   FALSE

(B)  f(x)= x+1

=> f(a+b) =  a+b + 1 
=> f(a) + f(b) =  a + 1  +  b + 1  =  a+b + 2  ≠  a+b + 1 
=> f(a+b) ≠ f(a) + f(b) =>   FALSE

(C)  f(x) = \sqrt{x}

=> f(a+b) =  \sqrt{a + b} 
 f(a) + f(b) =  \sqrt{a}  +  \sqrt{b}  ≠  \sqrt{a + b} 
=> f(a+b) ≠ f(a) + f(b) =>   FALSE

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

=> f(a+b) =  \frac{2}{a+b} 
 f(a) + f(b) =  \frac{2}{a}  +  \frac{2}{b}  =  \frac{2a + 2b }{ ab}  ≠  \frac{2}{a+b} 
=> f(a+b) ≠ f(a) + f(b) =>   FALSE

(E)  f(x) = -3x

=> f(a+b) =  -3*(a+b)  = -3a - 3b
 f(a) + f(b) =  -3a  +  -3b  = -3a - 3b
=> f(a+b) = f(a) + f(b) =>   TRUE

So,  Answer will be E 
Hope it helps!

Watch the following video to learn the Basics of Functions and Custom Characters

https://www.youtube.com/watch?v=b7JoMX0gk8M

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

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