1. Find a continuous function $\displaystyle f:(0,1) \rightarrow \mathbb {R} $ with an image equal to $\displaystyle \mathbb {R} $

I'm thinking may be something like arctanx + 1/2 ?

2. Find a continuous function $\displaystyle f:(0,1) \rightarrow \mathbb {R} $ with an image equal to [0,1]

tanx?

3. Find a continuous function $\displaystyle f: \mathbb {R} \rightarrow \mathbb {R} $ that is strictly increasing and has image equal to (-1,1).

Sinx might work but it ain't strictly increasing...