Given two functions whose outputs coincide at all values in the intersection of their domains, is there some commonly used notation to express the function whose graph is formed by the union of both those function's graphs?
Symbolically given any two functions $f:A\to X$ and $g:B\to Y$ with $f(r)=g(r)$ for all $r\in A\cap B$ I'm looking for some standard notation to write the function $h:A\cup B\to X\cup Y$ defined by:
$$h(r)=\begin{cases} f(r)& \text{if } r\in A\\ g(r)& \text{if } r\in B\end{cases}$$
So far I've just been writing $h=f\cup g$, as this notation seems pretty natural and also satisfies a lot of nice properties involving the standard usage of the union operator between sets. For example:
$$f\cup g=g\cup f$$ $$(f\cup g)\cup h=f\cup (g\cup h)$$ $$\text{dom}(f\cup g)=\text{dom}(f)\cup \text{dom}(g)$$ $$\text{range}(f\cup g)=\text{range}(f)\cup \text{range}(g)$$ $$(f\cup g)[U]=f[U\cap A]\cup g[U\cap B] \text{ for any subset } U\subseteq A\cup B$$ $$\text{ If } f \text{ and } g \text{ are bijective then so is } f\cup g \text{ further we have } (f\cup g)^{-1}=f^{-1}\cup g^{-1}$$
In particular a recent example of where this notation would have been useful, was when I was dealing with three functions $f:A\to X$ and $g:B\to Y$ and $h:C\to Z$ as well as two binary operations $*$ on $A\times B$ to $C$ and $\cdot$ on $X\times Y$ to $Z$ which satisfied:
$$h(a*b)=f(a)\cdot g(b)$$
Now using the previous notation this is equivalent to the identity:
$$(f\cup g\cup h)(a*b)=(f\cup g\cup h)(a)\cdot (f\cup g\cup h)(b)$$
Thus instead of writing all the aforementioned relations I could have just more concisely wrote that $f\cup g\cup h$ is a homomorphism from $(*,A\times B)$ to $(\cdot,X\times Y)$.
However for the sake of communicability I'm hesitant to define the "union" of functions in the way that I did and so I'm curious if there is some standard notation for expressing this binary operation between arbitrary functions.
In the usual ZFC formulation, a function is a set (or class) of ordered pairs (satisfying an additional axiom encoding uniqueness of function values). So if $f$ and $g$ coincide on $\mathrm{dom}(f) \cap \mathrm{dom}(g)$ then $f \cup g$ is a function on $\mathrm{dom}(f) \cup \mathrm{dom}(g)$, exactly the function $h$ you describe.
The notation is perfectly valid.