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

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

by melguy » Fri Jul 05, 2013 11:07 pm
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) = square root of x
F(x) = 2/x
F(x) = -3x

Please provide a quicker approach to this problem if any. I tried with picking numbers and it took me too long. Thanks
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by GMATGuruNY » Sat Jul 06, 2013 2:09 am
melguy wrote: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) = square root of x
F(x) = 2/x
F(x) = -3x

Please provide a quicker approach to this problem if any. I tried with picking numbers and it took me too long. Thanks
Plugging in values is an efficient approach to this problem.

Let a=2 and b=3.
f(a+b) = f(2+3) = f(5).
Question stem rephrased:

For which of the following functions does f(5) = f(2) + f(3)?

Answer choice A:
f(5) = 5² = 25
f(2) = 2² = 4
f(3) = 3² = 9
25 = 4+9 Doesn't work.

Answer choice B:
f(5) = 5+1 = 6
f(2) = 2+1 = 3
f(3) = 3+1 = 4
6 = 3+4 Doesn't work.

Answer choice C:
f(5) = √5
f(2) = √2
f(3) = √3
√5 = √2 + √3 Doesn't work.

Answer choice D:
f(5) = 2/5
f(2) = 2/2 = 1
f(3) = 2/3
2/5 = 1 + 2/3 Doesn't work.

The correct answer is E.

Answer choice E:
f(5) = -3*5 = -15
f(2) = -3*2 = -6
f(3) = -3*3 = -9
-15 = -6 + -9
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by alanforde800Maximus » Sat Jul 21, 2018 11:50 pm
GMATGuruNY wrote:
melguy wrote: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) = square root of x
F(x) = 2/x
F(x) = -3x

Please provide a quicker approach to this problem if any. I tried with picking numbers and it took me too long. Thanks
Plugging in values is an efficient approach to this problem.

Let a=2 and b=3.
f(a+b) = f(2+3) = f(5).
Question stem rephrased:

For which of the following functions does f(5) = f(2) + f(3)?

Answer choice A:
f(5) = 5² = 25
f(2) = 2² = 4
f(3) = 3² = 9
25 = 4+9 Doesn't work.

Answer choice B:
f(5) = 5+1 = 6
f(2) = 2+1 = 3
f(3) = 3+1 = 4
6 = 3+4 Doesn't work.

Answer choice C:
f(5) = √5
f(2) = √2
f(3) = √3
√5 = √2 + √3 Doesn't work.

Answer choice D:
f(5) = 2/5
f(2) = 2/2 = 1
f(3) = 2/3
2/5 = 1 + 2/3 Doesn't work.

The correct answer is E.

Answer choice E:
f(5) = -3*5 = -15
f(2) = -3*2 = -6
f(3) = -3*3 = -9
-15 = -6 + -9
Success!
Hello Mitch,

What if, we started directly with answer choice C, skipping choice A and B in first place as this question contains that phrase "which of the following"?
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by Scott@TargetTestPrep » Sun Aug 26, 2018 6:40 pm
melguy wrote: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) = square root of x
F(x) = 2/x
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 that answer choice E indeed 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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by Brent@GMATPrepNow » Mon Aug 27, 2018 5:29 am
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 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
Brent Hanneson - Creator of GMATPrepNow.com
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