We've updated our

TEXT

# Piecewise Linear Functions*

−x−3[/latex], erasing the part where x is greater than -3. Place an open circle at (-3,0). Now place the line $f(x) = x+3$ on the graph, starting at the point (-3,0). Note that for this portion of the graph, the point (-3,0) is included, so you can remove the open circle. The two graphs meet at the point (-3,0) The domain of this function is all real numbers because (-3,0)is not included as the endpoint of $f(x) = −x−3$, but it is included as the endpoint for $f(x) = x+3$. The range of this function starts at $f(x)=0$ and includes 0,  and goes to infinity, so we would write this as $x\ge0$[/hidden-answer] In the next example, we will graph a piecewise defined function that models the cost of shipping for an online comic book retailer.

### Example

An on-line comic book retailer charges shipping costs according to the following formula $S(n)=\begin{cases}1.5n+2.5\text{ if }1\le{n}\le14\\0\text{ if }n\ge15\end{cases}$ Draw a graph of the cost function.

Answer: First, draw the line $S(n)=1.5n+2.5$.  We can use transformations: this is a vertical stretch of the identity by a factor of 1.5, and a vertical shift by 2.5.

S(n)=1.5n+2.5
Now we can eliminate the portions of the graph that are not in the domain based on $1\le{n}\le14$
S(n) = 1.5n+2.5 for 1=n=14
Last, add the constant function S(n)=0 for inputs greater than or equal to 15. Place closed dots on the ends of the graph to indicate the inclusion of the end points.

In the following video we show how to graph a piecewise defined function which is linear over both domains. https://youtu.be/B1jfpiI-QQ8

## Summary

• A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.
• Evaluating a piecewise function means you need to pay close attention to the correct expression used for the given input
To graph piecewise functions, first identify where the domain is divided.  Graph functions on the domain using tools such as plotting points, or transformations. Be careful to use open or closed circles on the endpoints of each domain based on whether the endpoint is included.