perplexus.info :: Just Math : Positively Real

Positively Real (Posted on 2012-09-09)

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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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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