Piecewise Defined Functions

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 $x\le 2$, 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).