Define And Write Piecewise Functions | Intermediate Algebra

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Module 5: Linear Functions

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Define and Write Piecewise Functions

Learning Outcomes

• Define piecewise function
• Evaluate a piecewise function
• Write a piecewise function given an application

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

We use piecewise functions to describe situations where a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business where 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 where incomes up to [latex]$10,000[/latex] are taxed at [latex]10\%[/latex] and any additional income is taxed at [latex]20\%[/latex]. The tax on a total income, S, would be [latex]0.1[/latex]S if [latex]S\le[/latex] [latex]$10,000[/latex] and [latex]1000 + 0.2 (S - $10,000)[/latex] if S > [latex]$10,000[/latex].

Piecewise Function

A piecewise function is a function where more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all of these smaller domains. We notate this idea like this:

[latex]f\left(x\right)=\begin{cases}\text{formula 1 if x is in domain 1}\\ \text{formula 2 if x is in domain 2}\\ \text{formula 3 if x is in domain 3}\end{cases}[/latex]

In piecewise notation, the absolute value function is

[latex]|x|=\begin{cases}x\text{ if }x\ge 0\\ -x\text{ if }x