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 $f_k$.