# 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 $$y_2 = \frac{1}{2}x - 1$$: 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). This page titled 2.4: Algebra Tips and Tricks Part II (Piecewise Defined Functions) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.