find the function

[spinos]

Let the function \(\displaystyle f /(0,+\infty) \rightarrow (0,+\infty)\)
we know that:
• \(\displaystyle f(xy)= f[xf(\frac{1}{y})]\) for all x,y>0
•\(\displaystyle \lim_{x \to +\infty} f(x)=0\)

Find the \(\displaystyle f(x)\)

 

[Dr.Peterson]

Let the function \(\displaystyle f /(0,+\infty) \rightarrow (0,+\infty)\)
we know that:
• \(\displaystyle f(xy)= f[xf(\frac{1}{y})]\) for all x,y>0
•\(\displaystyle \lim_{x \to +\infty} f(x)=0\)

Find the \(\displaystyle f(x)\)

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We want to work with you in order to help, so we need to see what you have learned, what you have tried, and where you are stuck. You have given us none of that!

What techniques do you know for solving a problem like this? What have you discovered so far that might be helpful? Do you think we need to assume the function is continuous, or anything like that?

 

[spinos]

Yes, sorry for my rudeness.
I am math teacher in senior high school and this is a question of my student.
I teach to them function, limits , continuous, derivatives, integrals .

 

[spinos]

In begin the correct is:

[MATH]f: (0,+\infty) \rightarrow (0,+\infty) [/MATH]
I mean x>0 and f(x)>0

 

[Steven G]

You still haven't offered any type of solution.

 

[spinos]

And we don't know if f(x) is one to one (1-1) injective

 

[spinos]

\(\displaystyle f(x*y)= f(x*f(\frac{1}{y}))\) for all x, y >0

 

[pka]

I am math teacher in senior high school and this is a question of my student.
I teach to them function, limits , continuous, derivatives, integrals .

I really do not care for this question, particularly the notation.
BUT if \(f(t)=\dfrac{1}{t}\) then \(f\left(\dfrac{1}{y}\right)=y\)
Here is my objection to the notation: \(f(xy)=f\left(xf\left(\dfrac{1}{y}\right)\right)\)

 

[Dr.Peterson]

I am math teacher in senior high school and this is a question of my student.
I teach to them function, limits , continuous, derivatives, integrals .

It sounds like the problem is not directly related to the curriculum, but is perhaps a contest-type problem? I might help to know a little more about where it comes from.

I don't have a lot of knowledge about this sort of functional equation, but you imply that the problem doesn't assume such knowledge. So I'll just say what I would try first, since you've said nothing about what you have tried.

I would start by putting various specific values for x and y, and try to find potentially useful facts. We can't use 0, but try x=1, y=1, or x=1, y unspecified, and so on.

Please show what you find, or have already found.

EDIT: Although it doesn't seem helpful in proving uniqueness, it is often useful to just make some guesses; pka's suggestion may be more relevant than it initially looks.