Proving function is injective

Hey guys.

I've got the following problem:

Let og let be defined by :

(a) Show that *Done that already.*

(b) Show that is injective

(Hint: When is given by monotonic?)

(c)... some more questions.

I've already found the solution to (a), but I cant quite figure out how to solve (b). This is pretty much what I've tried so far:

For to be injective, the following has to be true : .

So, for proof by contradiction, we assume that for a pair we have that .

Since we have that (1).

But since we have that .

According to (1) we know that so must be the case.

From here I dont know exacly where to go. I can see that I'm not really using the hint.

I know that all monotonic functions are injective, but how can I prove that f is monotonic?

I can see by visualization that f will be monotonic, because x is defined to be positive, but how do I prove that rigorously?

Thanks for your help guys.

Morten

Re: Proving function is injective

the function is clearly continuous, since each component function is continuous. there is a theorem that says that a continuous real-valued function is injective if and only if it is strictly monotonic. but im not sure if there is an analogous theorem for

Re: Proving function is injective

Yes I understand that, but how do I prove that it is monotonic?