Finding a bijection and inverse to show there is a homeomorphism I need to find a bijection and inverse of the following:
$X = \{ (x,y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 4 \}$ with its subspace topology in $\mathbb{R}^2$ 
$Y = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1$ and $ 0 \leq z \leq 1 \}$ with its subspace topology in $\mathbb{R}^3$
Then show they are homeomorphic. Not sure where to start.

You can use the similar idea of Stereographic Projection. Take a point above the cylinder along the axix, say $P:=(0,0,2)$. Now join $P$ by a straight line to each point of the cylinder say $p$  and extend it till it hits the $x-y$ plane at point, say $p^\prime$. Then the map $p\mapsto p^\prime$ maps the cylinder to an annulus. Now scale that annulus properly to get your desired annulus.
