2.6 Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function [latex]\text{}f\left(x\right)=|x|\text{}[/latex]. With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude, or modulus, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

[latex]f\left(x\right)=x\text{}\text{ if }\text{}x\ge 0[/latex]

If we input a negative value, the output is the opposite of the input.

[latex]f\left(x\right)=-x\text{}\text{ if }\text{}x<0[/latex]

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. 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 [latex]\text{}S\text{}[/latex] would be [latex]\text{}0.1S\text{}[/latex] if [latex]\text{}S\le \text\$10\text{,}000\text{}[/latex] and [latex]\text{}\text\$1000+0.2\left(S-\text\$10\text{,}000\right)\text{}[/latex] if [latex]\text{}S>\text\$10\text{,}000[/latex].

Example 1: Writing a Piecewise Function

A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, [latex]\text{}n\text{}[/latex], to the cost, [latex]\text{}C[/latex].

Analysis

The function is represented in Figure 2-20. The graph is a diagonal line from [latex]\text{}n=0\text{}[/latex] to [latex]\text{}n=10\text{}[/latex] and a constant after that. In this example, the two formulas agree at the meeting point where [latex]\text{}n=10,\text{}[/latex] but not all piecewise functions have this property.

Solution

Example 2: Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, [latex]\text{}C,\text{}[/latex] in dollars for [latex]\text{}g\text{}[/latex] gigabytes of data transfer.

$$C(g)=\{\begin{array}{ccc}25& \text{if}& 0<g<2\\ 25+10(g-2)& \text{if}& g\ge 2\end{array}$$Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

Analysis

The function is represented in Figure 2-21. We can see where the function changes from a constant to a shifted and stretched identity at [latex]\text{}g=2\text{}[/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.

Solution

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