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
For which of the following functions is f(a+b) = F(a) + f(b)
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Plugging in values is an efficient approach to this problem.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
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!
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Hello Mitch,GMATGuruNY wrote:Plugging in values is an efficient approach to this problem.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
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!
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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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: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
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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One approach is to plug in numbers. Let's let a = 1 and b = 1For 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
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