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

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by DeepakR » Sun Apr 19, 2009 6:49 pm
Can someone please help me with the following attached question.
QA is E

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

1. f(x)=x^2
2. f(x)=x+1
3. f(x)=sqrt(x)
4. f(x)=2/x
5. f(x)=-3x

-Deepak
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by DanaJ » Sun Apr 19, 2009 10:32 pm
Basically eliminate functions. This can be done really quickly, it really takes more to try to write them on paper then solving it:

a. f(a + b) = (a + b)^2 = a^2 + b^2 + 2ab = f(a) + f(b) + 2ab - out

b. f(a + b) = a + b + 1 = a + 1 + b = f(a) + f(b) - 1 - out

c. f(a + b) = sqrt(a + b), which is almost always different from sqrt(a) + sqrt(b) - pick 9 and 16 to test that

d. f(a + b) = 2/(a + b), while f(a) + f(b) = 2/a + 2/b = 2(a + b)/ab - almost always different from 2/(a + b)

e. is the only one left:
f(a + b) = -3(a + b) = -3a - 3b = f(a) + f(b)
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by anthony.j » Mon May 04, 2009 7:23 am
Great explanation! Thanks!
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