# 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 [latex]f(x) = x+3[/latex] 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 [latex]f(x) = −x−3[/latex], but it is included as the endpoint for [latex]f(x) = x+3[/latex]. The range of this function starts at [latex]f(x)=0[/latex] and includes 0, and goes to infinity, so we would write this as [latex]x\ge0[/latex][/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 [latex]S(n)=\begin{cases}1.5n+2.5\text{ if }1\le{n}\le14\\0\text{ if }n\ge15\end{cases}[/latex] Draw a graph of the cost function.Answer: First, draw the line [latex]S(n)=1.5n+2.5[/latex]. 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 [latex]1\le{n}\le14[/latex]
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.
## 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

## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Determine a Basic Piecewise Defined Function.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Ex: Determine Function Values for a Piecewise Defined Function.
**Authored by:**James Sousa (Mathispower4u.com) .**License:**CC BY: Attribution. - College Algebra, Unit 1.4 Function Notation.
**Authored by:**Carl Stitz and Jeff Zeager.**Located at:**https://www.stitz-zeager.com/szca07042013.pdf.**License:**CC BY: Attribution. - Ex 2: Graph a Piecewise Defined Function.
**Authored by:**James Sousa (Mathispower4u.com) .**License:**CC BY: Attribution.

### CC licensed content, Specific attribution

- Precalculus.
**Provided by:**OpenStax**Authored by:**Jay Abramson, et al..**Located at:**https://openstax.org/books/precalculus/pages/1-introduction-to-functions.**License:**CC BY: Attribution.**License terms:**Download For Free at : http://cnx.org/contents/[email protected]..