It is well known that if $f:X\to Y$ is a morphism of affine varieties, then the image of $f$ is dense in $Y$ (ie $f$ is dominant) if and only if the homomorphism between coordinate rings, $f^*:K[Y]\to K[X]$, is injective.
Similarly, if $\phi:X\to Y$ is a dominant rational map, then $\phi^*:K[Y]\to K(X)$ is injective and hence can be extended to an (injective) homomorphism $\phi^*:K(Y)\to K(X)$.
However, I have not seen a converse to the latter in my textbook. What I have thought is that if there is a homomorphism between any two function fields, $\phi^*:K(Y)\to K(X)$, it must be injective automatically since they are fields, therefore it is not a matter of "when is it injective?", but rather one of "when does one exist?". However, any such $\phi^*$ clearly restricts to an injective homomorphism $\phi^*:K[Y]\to K[X]$, and we can conclude that $\phi^*$ induces a dominant morphism $\phi:X\to Y$ by the first statement.
My question is: is the last paragraph's argument correct? It seems 'obvious' but I have checked 4 different textbooks and none make any mention of it, so perhaps it is too obvious.