Classifying endomorphisms of $k[x]$ which are automorphisms

Question

Let $k$ be a field and let $\phi\colon k[x]\to k[x]$ be a $k$-linear ring homomorphism. I think the following is true but I am looking for a detailed proof or reference:

There exists an element $r\in k[x]$ satisfying $\phi(r)=x$ if and only if $\deg \phi(x)=1$.

EDIT:

I see this question was downvoted, perhaps I should add more context. This is not a homework question or anything of that sort - although if someone knows of a textbook or course that uses this as an exercise, just a reference to this would be enough to answer my question. As a secondary remark, assuming the result is correct, I am wondering under which conditions on a ring $R$ the result continues to hold with $R$ in place of $k$.

Answer 1

In fact, the following statements are all equivalent for any field $k$:

1. There is $r\in k[x]$ such that $\phi(r)=x$.
2. $\phi$ is surjective.
3. $\deg\phi(x)=1$.

Showing 1 to 2 and 3 to 1 is obvious, so I will leave it as your own.

Now assume that $\deg\phi(x)>1$. By observing the term of the highest order, we can see that $\phi(r)>0$ for any nonconstant $r\in k[x]$. Therefore, no $r\in k[x]$ satisfy $\phi(r)=x$, so $\phi$ is not surjective.

It is easy to see that $\phi$ is not surjective when $\deg\phi(x)<1$. Therefore, the surjectivity of $\phi$ implies $\deg\phi(x)=1$.