  Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let
$g:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists
of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup
E(G_{2})\cup \{uv:g(u)=v\}$. In this paper, we extend the study of the
distinguishing number of a graph to its functigraph. We discuss the behavior of
the distinguishing number in passing from $G$ to $F_{G}$ and find its sharp
lower and upper bounds. We also discuss the distinguishing number of
functigraphs of complete graphs and join graphs.
