find f satisfying f(f(x)) = -1/x, or show no such f exists

[littlegentleman]

I solved this question, but i am not sure about my answer.

Q. Is there a function f whose domain is the set of nonzero and real numbers and satisfies the identity f(f(x)) = -1/x? Show an example, or prove that there is no such function.

A. If f(f(x)) = -1/x, then f(f(x)) = -1/f(x)

Therefore, f(x) has to be [squareroot x]

For example, when x = 1, then f(x) = 1 and f(f(x)) = -1
and when x = -1, then f(x) = 1 and f(f(x)) = -1

Please tell me if I did something wrong.
Thank you very much.

 

[stapel]

A. If f(f(x)) = -1/x, then f(f(x)) = -1/f(x)

How did you arrive at this conclusion?

Therefore, f(x) has to be [squareroot x]

How did you arrive at this conclusion?

Note: If f(x) = sqrt[x], then f(f(x)) = sqrt[sqrt[x]] = x[sup:255s7n6k]1/4[/sup:255s7n6k], not -1/x. Also, sqrt[sqrt[1]] = 1, not -1, etc. So your proposed function cannot be correct.

Please reply with clarification, including a clear statement of your reasoning. Thank you!

Eliz.

Mixi and Art of Problem Solving

 

[littlegentleman]

Sorry, I totally misunderstood the question.
I read the question more time and now i get it.
My new answer is that There is no such function. Because if f(x) is negative, then it will become positive after f(f(x)).
And if it is positive, then it will stay positive after f(f(x)). It is not possible.
Is this correct?
Please tell me if i did something wrong.
Thank you very much.

 

[stapel]

There is no such function. Because if f(x) is negative, then it will become positive after f(f(x)).
And if it is positive, then it will stay positive after f(f(x)). It is not possible.

I'm sorry, but I don't understand what you mean...?

How did you conclude that, if f(x) were negative, then f(f(x)) would be positive? For instance, if f(x) = -| x |, then f(f(x)) = -|-|x||, which is still negative.

Also, if f(f(x)) has the opposite sign if f(x) is negative, then why would f(f(x) have the same sign if f(x) is positive?

Please reply with your reasoning and justification. Thank you!

Eliz.

 

[littlegentleman]

I don't know how to explain this in words and i can't find any mathematical way to solve this :?
But i come up with an ansewr -1/ IxI. Therefore, f(f(x)) = -1/I -1/IxII = -1 / x
IIxII = x? is this right? :? :? :?
Please tell me if i did something wrong. Well, this one is tough.
Thank you very much.

 

[stapel]

i come up with an ansewr -1/ IxI.

If f(x) = -1/|x|, then:

. . . . .f(f(x)) = -1/[f(x)] = -1 / [ -1 / | x | ] = (-1 / 1) (| x | / -1) = | x |

...not -1/x, as required. :shock:

Eliz.

 

[littlegentleman]

well...i have no idea how to solve this one...
All i can do is guessing and checking if that is correct.
Can you show me the answer now if that's possible?
I really appreciate that.
Thank you very much.

 

[stapel]

Try the links below my signature in my first reply. :wink:

Eliz.