Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 230

Find all functions

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

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

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).

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.

tony123

Member

Registered: 2007-08-03

Posts: 230

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)

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.