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

[rsarashi]

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

A) f(x) = x²

B) f(x) = x+1

C) f(x) = under root x

D) f(x) = 2/x

E) f(x) = -3x

OAE[/spoiler]

[GMATGuruNY]

I posted a solution here:
https://www.beatthegmat.com/for-which-of ... 65920.html

[Brent@GMATPrepNow]

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

f(x)= x²
f(x)= x+1
f(x)= √x
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)=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

[[email protected]]

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

[Jeff@TargetTestPrep]

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

A) f(x) = x²

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) = √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) = 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