prove there exists an inverse function

Hello. I'm quite new to calculus and frankly speaking I'm an autodidact. Could you tell me how to solve this:

Prove that there exists a inverese of . In what set is differentiable? Calculate

I would really appreciate a thorough explanation. Thank you.

Re: prove there exists an inverse function

Well, you could easily show that on the interval , f is monotonously increasing, since exists on the interval, it is continuous on the interval. A continous function which is monotously increasing or decreasing is one to one on that interval and thus also has an inverse. Now if f is differentiable on the interval, is also differentiable given that at any point on the interval. You can verify that this is true. Thus is differentiable over