# Glossary for Lesson 23: Piecewise and Rational Functions

### Piecewise Function

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 f is paired with a certain part of the domain of f .  Graphs of piecewise functions sometimes end up looking really interesting.  Here, for example, is the graph of the function shown above:

### Step Function

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

### Rational Function

A rational function is any function that is a ratio of two polynomials. That is, a rational function is of the form \frac {p}{q} , where p and q are both polynomials and q\neq 0 . Here are some examples of rational functions:
y=\frac {3x-1}{x^2+6x+8}
y=\frac {1}{x}
y=\frac {4x^3-9x+12}{x+8}
y=\frac {(x+4)(x-2)}{x(5x+2)(x-2)}
All polynomial functions are technically rational functions, since any polynomial p(x) can be written as \frac {p(x)}{1} , and 1 is a polynomial of order zero.