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#1 2008-07-20 18:38:52

tony123
Member
Registered: 2007-08-03
Posts: 189

Find all functions

Find all functions f: (0,∞)------ (0,∞) such that



f(f(f(x))) + 4f(f(x)) + f(x) = 6x .

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#2 2008-07-20 21:23:02

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Find all functions

I suspect f has to be linear. If so, let

, and you’ll find that there are only three possibilities:
,
,
.

I don’t know if f can be non-linear (though I have a hunch that it can’t).


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#3 2008-07-22 07:19:53

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Find all functions

Also

So f must be injective. That’s another thing we can say about the function.


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#4 2008-07-23 04:40:36

tony123
Member
Registered: 2007-08-03
Posts: 189

Re: Find all functions

thank you jane

but i think The only solution of function is f(x) = x,

what can be easily checked.

Last edited by tony123 (2008-07-23 04:49:41)

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#5 2008-07-23 05:09:31

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Find all functions

You’re right. I just noticed that f is from

to
rather than
to
, so
and
will not work.


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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