A *piecewise function* is a function that is made up of pieces of different functions in different parts of its domain. We could define a piecewise function like this:

Notice here that each part, or each so-called *subfunction*, of the piecewise function is paired with a certain part of the domain of . Graphs of piecewise functions sometimes end up looking really interesting. Here, for example, is the graph of the function shown above:

A *step function* is a piecewise function whose parts are constant functions, or horizontal lines. A step function might look something like this:

where the equation of the function is

A *rational function* is any function that is a ratio of two polynomials. That is, a rational function is of the form , where and are both polynomials and .
Here are some examples of rational functions:

All polynomial functions are technically rational functions, since any polynomial can be written as , and is a polynomial of order zero.