Finding f(6) from a composite function

[Akash47]

Homework Statement

If x, a, b are positive integers, f(x) is positive integer too. And if a>b, then
f(a)>f(b).What is f(6)?

Relevant Equations

f(f(x))= x^2+2

It is obvious that the function f is not injective. From the given equation, we get f(f(2))=6.And since,there is an inequality given in the problem, I think we can use that to find f(6).But I have got stuck here and can't move.Do I have to find what is f(x) first?Then how?

 

[SammyS]

It is obvious that the function f is not injective. From the given equation, we get f(f(2))=6.And since,there is an inequality given in the problem, I think we can use that to find f(6).But I have got stuck here and can't move.Do I have to find what is f(x) first?Then how?

If you consider that for this exercise, f does not need to be defined for negative numbers, then f can be treated as being injective. However, I don't see that being injective would help.

I don't see how to find f(x) in general.

Use the given information to find some possible values for f.

Start with f(1).
What do you know? f(1) is a positive integer. Also, you know that f(f(1)) = 3 .

You also know that f(1) < f(2) < f(3) < ... : Right?

 

[Akash47]

I know that f(1) < f(2) < f(3) , then f(f(1)) <f(f(2)) <f(f(3)) and that implies, 3<6<11! Who doesn't know this?I had tried this method and that didn't help me that's why I have posted the problem? Can you please help me a little bit more?

 

[Dick]

I know that f(1) < f(2) < f(3) , then f(f(1)) <f(f(2)) <f(f(3)) and that implies, 3<6<11! Who doesn't know this?I had tried this method and that didn't help me that's why I have posted the problem? Can you please help me a little bit more?

Use that $(f\circ f) \circ f == f\circ (f \circ f)$ (where $\circ$ is function composition) to conclude that $f(x^2+2)=f(x)^2+2$. That let's you get more relations. SammyS is suggesting you guess values for $f(1)$ and see if they lead to contradictions. Try it. For example, can $f(1)=1$?

 

[DaveE]

Try making a table with columns of x, f(x), f(f(x)). Then start with some guesses to start out the f(x) column. You can then infer other table entries resulting from your guess and rule out the ones that violate the given restrictions. Similar to what you did to find 3<6<11.
I'm with SammyS; I don't have a clue how to find a closed form answer to what f(x) is. But I do know what f(6) is.

 

[Akash47]

Thanks to all.I have solved the problem.

 

[SammyS]

Thanks to all. I have solved the problem.

It's nice to know that you solved the problem.

However, the people who help students gain understanding and skills to solve problems here at PF are volunteers .

Our only reward is knowing that you have been helped, so, yes it's good that you indicted that you have a solution. A more satisfying reward for us is to actually see that solution, particularly the details we may have helped you with.