6. For which of the following functions is f(a+b)=f(a)+f(b)?
A) f(x) = x2
B) f(x) = x+1
C) f(x) = root x
D) f(x) = 2/x
E) f(x) = -3x
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GMAT Prep Question # 3
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Brent@GMATPrepNow
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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)
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.
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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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We will have to solve for each option in this problem.
We can assume numbers or can just go ahead with a and b.
As Brent has already taken the assumption method, I would try to explain it by a and b
Option A:
f(a) = a^2, f(b) = b^2, so f(a)+f(b) = a^2+ b^2
f(a+b) = (a+b)^2 = a^2 + b^2 +2ab
This is not equal to f(a) + f(b)
So, we can rule out option A
Option B:
f(a)= a+1, f(b) = b+1, so f(a)+f(b) = a+b+2
f(a+b) = a+b+1
This is not equal to f(a) + f(b)
So, we can rule out option B
Option C:
f(a) = (a)^1/2, f(b) = (b)^1/2. So f(a)+f(b) = (a)^1/2+ (b)^1/2
f(a+b) = (a+b)^1/2
This is not equal to f(a) + f(b)
So, we can rule out option C
Option D:
f(a) = 2/a, f(b) = 2/b. So f(a)+f(b) = 2/a + 2/b = 2(a+b)/ab
f(a+b) = 2/(a+)
This is not equal to f(a) + f(b)
So, we can rule out option D
Hence, we are left with option E as the answer.
Still for confirmation, let us check it also
Option E:
f(a) = -3a, f(b) = -3b. So f(a)+f(b) = -3a + (-3b) = -3(a+b)
f(a+b) = -3(a+b)
This is equal to f(a) + f(b)
Hence E is the correct option.
We can assume numbers or can just go ahead with a and b.
As Brent has already taken the assumption method, I would try to explain it by a and b
Option A:
f(a) = a^2, f(b) = b^2, so f(a)+f(b) = a^2+ b^2
f(a+b) = (a+b)^2 = a^2 + b^2 +2ab
This is not equal to f(a) + f(b)
So, we can rule out option A
Option B:
f(a)= a+1, f(b) = b+1, so f(a)+f(b) = a+b+2
f(a+b) = a+b+1
This is not equal to f(a) + f(b)
So, we can rule out option B
Option C:
f(a) = (a)^1/2, f(b) = (b)^1/2. So f(a)+f(b) = (a)^1/2+ (b)^1/2
f(a+b) = (a+b)^1/2
This is not equal to f(a) + f(b)
So, we can rule out option C
Option D:
f(a) = 2/a, f(b) = 2/b. So f(a)+f(b) = 2/a + 2/b = 2(a+b)/ab
f(a+b) = 2/(a+)
This is not equal to f(a) + f(b)
So, we can rule out option D
Hence, we are left with option E as the answer.
Still for confirmation, let us check it also
Option E:
f(a) = -3a, f(b) = -3b. So f(a)+f(b) = -3a + (-3b) = -3(a+b)
f(a+b) = -3(a+b)
This is equal to f(a) + f(b)
Hence E is the correct option.
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nikhilgmat31
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Brent's solution is easy .
Initially I was surprised & think that something is missing in question
I got the question. Thanks Brent.
Initially I was surprised & think that something is missing in question
I got the question. Thanks Brent.
