Suppose $f_1,f_2, ... ,f_n$ are real valued functions with disjoint domains ( subsets of  real numbers): $D_1,D_2, ... ,D_n$.
Let $D = D_1 \cup D_2 \cup ... \cup D_n$, the union of the disjoint domains, so that $x \in D$ if and only if there is exactly one $k \in \{1,2,...,n\}$ where $x \in D_k$.

Definition: We say that $f$ 
is a function defined by (piecewise) cases on the domain $D$ if for any $x \in D$, with $ x \in D_k$ the value of $f(x)= f_k(x)$.
Comment: The cases are frequently announced in the definition of $f$ by statements like "$f(x) = ....$ if $ x$ satisfies some condition" where the conditions are mutually exclusive.
This can be visualized on a mapping diagram by highlighting the distinct subsets $D_k$ and labeling arrows with the appropriate function name $f_k$.