I have searched, but I haven't found an answer to this question. How are piecewise functions defined in set theory? And what are alternative ways to represent them?

Let's say we have a piecewise function $f:\mathbb{R}\_+ \to \mathbb{R}$, that is defined by the equation

$$ f(x)=
 \begin{cases} 
 x & x \in [0,3] \\
 3 & x \in (3,8] \\
 x^2 & x \in (8,\infty) 
\end{cases}
$$

The graph of this function is the set 

$$ \text{graph } f= \{ (x,y) \mid y=f(x), x \in \mathbb{R}\_+ \}, $$

but how can I write out this set "graph $f$" using information from the above equation? My quess would be something like this:

$$ \{ (x,y) \mid (x\in[0,3] \rightarrow y=x) \text{ or } (x \in (3,8] \rightarrow y=3) \text{ or } (x \in (8,\infty) \rightarrow y=x^2) \} $$

Is there anything else to know about piecewise functions in set theory, other than the function's graph?