Two different maps of a cylinder into $\Bbb R^3$

Question

Suppose two maps from a cylinder into $\Bbb R^3$ :

$f : [0,1] \times S^1 \to \Bbb R^3 $ given by $(t, \theta) \mapsto (\cos\theta, \sin\theta, t)$,

$g : [0,1] \times S^1 \to \Bbb R^3 $ given by $(t, \theta) \mapsto (\cos\theta, \sin\theta, 0)+$$ t\over{4} $$(\sin\theta \cos \theta, \sin\theta \sin \theta, \cos \theta) $.

What I want to show are the followings :

1. Is $g$ is an embedding?
2. Is there a homeomorphism $\phi$ of $\Bbb R^3$ such that $\phi ~\circ f=g $?

(I think this question is related to homotopy and covering space theory so I tagged these, but actually I'm not sure about this.) Any hints?

Answer 1

This is not an answer but a visual hint that can help you solve the problem. Below the plot of $g$ (educated guess that $S^1$ is already parametrized ashenno mentioned in a comment).

$g$"> The plot of $f$ is ]2 The hint is to look at the boundary of the surfaces and think of how they are linked. The boundary of $g$ is This is how the boundary of $f$ looks like: (The boundaries are fattened so you can see them)