2.4: Algebra Tips and Tricks Part II (Piecewise Defined Functions)

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Piecewise Defined Functions

Graph the following function

$g(x) = \left\{ \begin{array}{lr} x-1 & x \leq 2 \\ \frac{1}{2}x- 1 & x > 2 \end{array} \right.$

How do you do it? Well, you have to graph two different lines: $y_1 = x-1$ and $y_2 = \frac{1}{2}x - 1$:
$Shows two lines for the purpose of graphing a piecewise-defined function$

But then you need to “cut off” the graph of $y_1$ after $x = 2$, and “cut off” the graph of $y_2$ before $x = 2$:

$Take the previous image and cuts off each line in the appropriate spot to graph the piecewise defined function$

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).

$Shows filled-in dots represents the endpoint is included, and an open dot represents the opposite.$