2.4: Algebra Tips and Tricks Part II (Piecewise Defined Functions)
Piecewise Defined Functions
\(g(x) = \left\{ \begin{array}{lr} x-1 & x \leq 2 \\ \frac{1}{2}x- 1 & x > 2 \end{array} \right.\)
How do you do it? Well, you have to graph two different lines: \(y_1 = x-1\) and \(y_2 = \frac{1}{2}x - 1\):
But then you need to “cut off” the graph of \(y_1\) after \(x = 2\), and “cut off” the graph of \(y_2\) before \(x = 2\):
That’s the graph of \(g(x)\)! It is called a piecewise defined function . Since each piece is linear, sometimes it is called a piecewise linear function .
There is one more detail to clear up. What is the value of the function at \(x=2\)?
Well, going back to the original function, we see that \(g(x)\) was defined as \(x-1\) for \(x \leq 2\), and this includes \(x=2\). So we should use the blue line to determine the y-coordinate for \(x=2\). To indicate this on the graph, a filled in dot can be added to the blue graph (indicating the endpoint is included), and an open or not-filled-in dot is added to the green graph (to indicate the endpoint is not included).