Let $f: \Bbb{Q}[[x]]\rightarrow\Bbb{Q}[[x]]$ be a $\Bbb{Q}$ homomorphism of algebras and $f$ sends every inverse element in inverse elements.
a)Show that $f(\langle x \rangle)\subset \langle x \rangle $
b) Is that true also for powers $\langle x \rangle^k$
Proof: a) If we take an element $h$ in $\langle x \rangle$ and say that $g=f(h)$ is not in $\langle x \rangle$, then $g(0)=c \in \Bbb{Q}^*$ and $h-c\in \Bbb{Q}[[x]]^*$. But then $f(h-c)=g-g(0)\in\langle x \rangle$ would be not invertible. Proof: b)Yes, since $\langle x \rangle^k=\langle x^k \rangle$ and $f(\langle x \rangle^k)=\langle f(x^k) \rangle=\langle f(x)^k \rangle \in \langle x \rangle^k$.
Why from $g=f(h)$ is not in $\langle x \rangle$, follows $g(0)=c \in \Bbb{Q}^*$ and $h-c\in \Bbb{Q}[[x]]^*$ and $f(h-c)$ would not be invertible in a) And in b: could somebody please explain me the equalities, does it follow from properties of functions or polynomials?
Thanks in advance for the help