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.