Piecewise-Defined Function
A piecewise-defined function is one which is defined not by a single equation, but by two or more. Each equation is valid for some interval .
Example 1:
Consider the function defined as follows.
The function in this example is piecewise-linear, because each of the three parts of the graph is a line.
Piecewise-defined functions can also have discontinuities ("jumps"). The function in the example below has discontinuities at and .
Example 2:
Graph the function defined as shown.
Note that we use small white circles in the graph to indicate that the endpoint of a curve is not included in the graph, and solid dots to indicate endpoints that are included.
Example 3:
Graph the function defined below.
Negative values of and are not included in the domain because the first function, , is undefined for those values. The value is not included in the domain because the second function is not defined for that value (it has a vertical asymptote there). Therefore the domain of this function is . This can be represented using interval notation as .