"Piecewise" or "Case by case" Functions
SignificanceAll of the functions in the field guide can be added, multiplied, subtracted and divide (with reasonable precautions against dividing by zero). They can be composed (if domains and co-domains match up). They can also be combined to make functions whose formulas are different on different parts of the domain. These are called piecewise functions and are useful as examples as well as for representing behavior whose governing law switches at some point in the domain.
Standard NotationThe notation
Often piecewise functions have three or more separate functions and conditions. To see check that we understand how the conditions work, let's look at the function
f(-2) = (-2)2 = 4 because -2 is in the interval where the f(x)=x2
Graphing Piecewise FunctionsWhen graphing piecewise functions it is important to indicate what happens at the ends of the pieces. In the above example, the point (-1,1), which lies on the graph of y = x2, is NOT included in the graph of y = f(x). this is indicated on the graph by an empty circle. The point (-1,-1), on the other hand, IS included. This is indicated by a solid circle. An alternative method is to use square and round brackets as one would to indicate open and closed intervals. This is illustrated on the right.