Let us work in ZFC set theory, let A and B be two sets and C be the set of functions with domain A and range B.
Question: what can be said about c=rank(C), knowing a=rank(A) and b=rank(B) ?
Gérard Lang



About the rank of sets
Note that every function $f$ is a subset of $A\times B$, and so the rank of $C$ is at most $\newcommand{\rank}{\operatorname{rank}}\rank(\mathcal P(A\times B))$.


It is possible that $\rank(A\times B)=\rank(A)=\rank(B)$. For example in the case $A=B=\omega$. But it is possible that $\rank(A\times B)>\rank(A),\rank(B)$. For example where $A$ and $B$ are finite sets.


More generally if $\max\{\rank(A),\rank(B)\}$ is a limit ordinal then you can show that $A\times B$ will have the same rank as that maximum, otherwise it will have a strictly larger rank.


To be on the safe side, you can always show that $\rank{A^B}\geq\max\{\rank(A),\rank(B)\}+3$.


