Piecewise Functions

A piecewise function is a function which is defined by multiple subfunctions, each subfunction applying to a certain interval of the main function's domain (a subdomain). Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. For example, a piecewise polynomial functions: a function that is a polynomial on each of its subdomains, but possibly a different one on each.

Note that, in this case, we have three different “formulas” for f(x): -x + 1, 2 and x²- 4. But remember, with a function, when you evaluate f(x), you should get ONE ANSWER—not three. Otherwise, you don’t have a function.

How do you know which answer you should have (i.e., which formula you should use)? The “formula” is chosen using the right-side of the piecewise-defined function (the “if …” part).

For example, suppose we wanted to find f( − 3). Since − 3 < − 1, we will use the first “formula”:

f( − 3) = - (-3) +1 = 4.

If we want to find f(5), since 5 > 3, we use the third “formula”:

f(5) = 5² - 4 = 21.

What about finding f(1)? Note that in the interval (-1,3) the value of function is 2.

The following applet shows the graph of piece wise function 

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com