Some functions come in pieces. In this section, we will learn about how to define and graph functions that are essentially collections of discrete pieces.  Examples of something defined this way include designing the profile of a car, figuring out your mobile phone plan, and calculating income tax rates.  For example, your tax rate depends on your income and is the same for a range of incomes, as is shown in the table below:

A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income, S, would be 0.1S if [latex]S\le[/latex] $10,000 and 1000 + 0.2 (S – $10,000), if S> $10,000.

Evaluate a Piecewise Defined Function

In the first example, we will show how to evaluate a piecewise defined function.  Note how it is important to pay attention to the domain to determine which expression to use to evaluate the input.

In the following video we show how to evaluate several values given a piecewise defined function.

In the next example we show how to evaluate a function that models the cost of data transfer for a phone company.

Analysis of the Solution

The function is represented in the graph below. We can see where the function changes from a constant to a line with a positive slope at [latex]g=2[/latex]. We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.

C(g) = [latex]C\left(g\right)=\begin{cases}{25}\text{ if }{ 0 }<{ g }<{ 2 }\\ 10g+5\text{ if }{ g}\ge{ 2 }\end{cases}[/latex]

 Write a Piecewise Defined Function

In the last example we will show how to write a piecewise defined function that models the price of a guided museum tour.

Analysis of the Solution

The function is represented in Figure 21. The graph is a diagonal line from [latex]n=0[/latex] to [latex]n=10[/latex] and a constant after that. In this example, the two formulas agree at the meeting point where [latex]n=10[/latex], but not all piecewise functions have this property.

In the following video we show an example of writing a piecewise defined function given a scenario.

Graph Piecewise Functions

In this section, we will plot piecewise functions. The function plotted below represents the cost to transfer data for a given cell phone company. We can see where the function changes from a constant to a line with a positive slope at [latex]g=2[/latex]. When we plot piecewise functions, it is important to make sure each formula is applied on its proper domain.[latex]C\left(g\right)=\begin{cases}{25} \text{ if }{ 0 }<{ g }<{ 2 }\\10g+5\text{ if }{ g}\ge{ 2 }\end{cases}[/latex]

In this case, the output is 25 for any input between 0 and 2.  For values equal to or greater than 2, the output is defined as [latex]10g+5[/latex].

Example

Sketch a graph of the function.

Given the piecewise definition [latex]f(x)=\begin{cases}−x − 3\text{ if }x < −3\\ x + 3\text{ if } x \ge −3\end{cases}[/latex]
Draw the graph of f.
State the domain and range of the function.

Show Answer

First, graph the line [latex]f(x) = −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]

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.

Show 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.

In the following video we show how to graph a piecewise defined function which is linear over both domains.

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.