I am self studying Field Theory from Thomas Hungerford and I have a question in the proof of this theorem. 
In line 6th of proof, why there shouldn't be $F = π(K(x))$ isomorphic to $π(K[x])$ isomorphic to $K[x]$ isomorphic to $K(x)$ . Because $π: K[x] to K[x]/(f)$.
But why author writes "$F$ contains $π(K)$ isomorphic to $K$".
(2) Also, in 2 last line I am unable to think about why this side $F$ is subset of $K(u)$ must hold. ( For proving $F=K(u)$ ).
Can someone please tell where I am missing something.
For the first question, the restriction of $\pi$ to $K$ $\pi|_{K}$, is a monomorphism, so its restriction to its image $\hat{\pi}|_K:K \rightarrow \operatorname{Im}( \pi|_K)$ is an isomorphism and thus $K \cong \operatorname{Im}(\pi|_K) $. Clearly $\operatorname{Im}(\pi|_K) \subset K[x]/(f)=F$, so we conclude the result.
For the second one, let $u=\pi(x)=x+(f(x)) \in F$. An element of $F$ will be of the form $a_0+a_1x+...+a_nx^n + (f(x))$, with $a_0,\dotsc,a_n \in K$. We have:
$$ a_0+a_1x+ \dotsc+a_nx^n + (f(x)) \\ =(a_0+(f(x)))+(a_1+(f(x)))(x+(f(x)))+\dotsc+(a_n +(f(x)))(x+(f(x)))^n \\ =(a_0+(f(x)))+(a_1+(f(x)))u+\dotsc+(a_n +(f(x)))u^n \in K(u) $$
We then conclude $F \subset K(u)$.