The steps are to divide the domain x into intervals. Each interval will be a function of x accompanied by a range of x vales for which the function applies. The intervals may overlap to provide continuity, but usually one interval starts immediately after the previous one enters for each interval. The function doesn't have to be continuous but there should be a complete set of intervals to cover x from -∞ to +∞.
The function is written: f(x)=
{ f1(x) | x<a1
{ f2(x) | a1≤x<a2
{ ...
{ fn-1(x) | an-2≤x<an-1
{ fn(x) | x≥an-1 where f1 to fn represent different functions of x, n is the number of "pieces", a1 to an-1 represent constants in ascending size order so that the whole domain is defined. The vertical bar stands for the word "for". Each inequality shows an interval over which the function applies. For example, a3≤x<a4 means x in the interval [a3,a4) (open-ended). In other words, if we give values to the constants, we could have [0,3) meaning 0≤x<3, that is, x is between 0 (inclusive) and 3 (excluding x=3). The next interval would define the function between 3 (inclusive) and the next constant.
For example: f(x)=
{ -1 | x<0
{ 2x-1 | 0≤x<3
{ 5 | x≥3