simple function composite question

and are functions from to

1) If is a monotonically increasing function, does that mean that is monotonically increasing function?

2) If and are monotonically increasing functions, does that mean that is a monotonically increasing function

I think for the first one must be a monotonically increasing funtion. It doesn't matter what is since it is in the domain of . We can define however we want to but if is a monotonically increasing function, then definitely is too.

The second one I think is false for the reasons I just gave.

Am I correct?

Thanks!