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Positively Real (Posted on 2012-09-09) Difficulty: 3 of 5
R+denotes the set of positive real numbers.

Determine all possible functions f:R+ R+ such that:

f(f(x)) = 12x - f(x) for all x > 0.

No Solution Yet Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Some Thoughts Possible solution | Comment 2 of 4 |
Assume f(x) = ax
Then a*ax = 12x - ax
        a = 3 or -4
but f(x) = -4x does not map positives to positives.  Thus, a must = 3.

Assume f(x) = b/x
Then x = 12x - b/x
There is no value of b such that this equation works for all x. Therefore, f(x) is not of the form b/x.

I think that the same is true if f(x) = ax^b where b <> 1

I have not exhausted all functions that map R+ R+ ,
but I suspect that the only solution is f(x) = 3x

  Posted by Steve Herman on 2012-09-09 20:21:44
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