Lang problem about automorphism of polynomial ring STATEMENT: Let $A$ be a commutative entire(integral domain) ring and $X$ a variable over $A$. Let $a,b\in A$ and assume that $a$ is a unit in $A$. Show that the map $X\mapsto aX+b$ induces a unique automorphism of $A[X]$ inducing the identity on $A$. What is the inverse map?

QUESTION: Not really sure how to show the surjective part of the mapping to finish proof that it is an isomorphic map. Some help would be appreciated.
It is not too difficult to guess the inverse map. Since it is a twosided inverse, we get injectivity and surjectivity for free from its existence.
It is not too difficult to guess the inverse map. Since it is a twosided inverse, we get injectivity and surjectivity for free from its existence.