How to prove that this function is bijective...

Hi, how can I prove that , where

is a bijective function?

If the function were just , the problem could be easily solved by noting that and are strictly increasing functions, and their linear combination is therefore also strictly increasing. As the image of is whole , is bijective.

But in , we also have , which is a periodic, and not strictly monotonous function. Do we have to show that is simultaneously injective and surjective to be a bijection or is there some other way?

Thanks!