BigUp wrote:

Hello GmatClub,

Could someone please help me to understand this gmatprep question? Thanks!

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)=sqrt(x)

f(x)=2/x

f(x)=-3x

First, a remark: \(a\) and \(b\) are considered positive because the function in C, the square root is not defined for negative numbers and the function in D is not defined for \(x=0\) (\(x\) being in the denominator). The other functions are defined for any real number.

For each function, we translate the given equality and check whether is holds for any positive \(a\) and \(b\). If the equality holds for any \(a\) and \(b\), we should get an identity, which means the same expression on both sides of the equal sign.

For \(f(a+b)\) we take the expression of any of the given functions and replace \(x\) by \((a+b)\).

(A) \((a+b)^2=a^2+b^2\) or \(a^2+2ab+b^2=a^2+b^2\). Necessarily \(ab=0\), which cannot hold, \(a\) an \(b\) being positive. NO

(B) \(a+b+1=a+1+b+1\) gives \(1=2\), impossible. NO

(C) \(\sqrt{a+b}=\sqrt{a}+\sqrt{b}\) NO (check for example \(a=b=1\))

(D) \(\frac{2}{a+b}=\frac{2}{b}+\frac{2}{b}\) NO (again, check for \(a=b=1\))

(E) \(-3(a+b)=-3a+(-3b)\) or \(-3a-3b=-3a-3b\) YES!!!

Answer E.

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PhD in Applied Mathematics

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