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

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rsarashi: Master | Next Rank: 500 Posts; Posts: 186; Joined: Sat Dec 24, 2016 12:38 amFollowed by: 3 members; Thanked: 5 times

by rsarashi » Thu Jul 27, 2017 9:37 am

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

A) f(x) = xÂ²

B) f(x) = x+1

C) f(x) = under root x

D) f(x) = 2/x

E) f(x) = -3x

OAE[/spoiler]

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Source: Beat The GMAT — Problem Solving | Thu Jul 27, 2017 9:37 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Jul 27, 2017 9:49 am

I posted a solution here:
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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Jul 27, 2017 11:27 am

For which of the following functions f(a+b) = f(a) + f(b) for all positive numbers a and b?

f(x)= xÂ²
f(x)= x+1
f(x)= âˆšx
f(x)= 2/x
f(x)= -3x

One approach is to let a = 1 and b = 1 and plug in the values.

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)=xÂ², does f(2) = f(1) + f(1)?
Plug in to get: 2Â² = 1Â² + 1Â²? (No, doesn't work)
So, it is not the case that f(2) = f(1) + f(1), when f(x)=xÂ²

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

Brent Hanneson - Creator of GMATPrepNow.com

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Thu Jul 27, 2017 11:40 am

Hi rsarashi,

This question can be solved by TESTing VALUES. When dealing with these types of questions, you essentially have to perform 3 calculations on each answer: an end result when X=A, a result when X=B and a result when X=(A+B).

For example: When X=1, when X=4 and when X =(4+1)=5

We're looking for the function that will end with the sum of the first two calculations equaling the third calculation.

Answer A: f(X) = X^2
1^1 = 1
4^2 = 16
5^2 = 25
1+16 does NOT equal 25, so this is NOT the answer.

Answer B: f(X) = X+1
1+1 = 2
4+2 = 6
5+2 = 7
2+6 does NOT equal 7, so this is NOT the answer.

Answer C: f(X) = X^(1/2)
1^(1/2) = 1
4^(1/2) = 2
5^(1/2) = about 2.2
1+2 does NOT equal 2.2, so this is NOT the answer.

Answer D: f(X) = 2/X
2/1 = 2
2/4 = .5
2/5 = .4
2+.5 does NOT equal .4, so this is NOT the answer.

Answer E: f(X) = -3X
(-3)(1) = -3
(-3)(4) = -12
(-3)(5) = -15
(-3)+(-12) DOES equal -15, so (since we eliminated the other 4 answers) this MUST be the answer.

Final Answer: E

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 amFollowed by: 22 members; Location: New York, NY; Thanked: 39 times

by Jeff@TargetTestPrep » Wed Aug 09, 2017 12:43 pm

rsarashi 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) = xÂ²

B) f(x) = x+1

C) f(x) = under root x

D) f(x) = 2/x

E) f(x) = -3x

OAE[/spoiler]

We need to determine when f(a + b) = f(a) + f(b). We can determine the correct answer choice by substituting numerical values for a and b. We could use any two values for a and b, but for simplicity, let's choose a = 1 and b = 2. The function now looks like this:

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

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

So, we must determine which answer choice(s) has f(3) equal to the sum of f(1) and f(2).

Let's 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 3 + 2, choice B is not correct.

C) f(x) = âˆšx

f(3) = âˆš3

f(1) = âˆš1 = 1

f(2) = âˆš2

Since âˆš3 does not equal 1 + âˆš2, choice C is not correct.

D) f(x) = 2/x

f(3) = 2/3

f(1) = 2/1 = 2

f(2) = 2/2 = 1

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

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

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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