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Section 2.2 Piecewise Defined Functions Activity

Subsection 2.2.1 Evaluating Pieces of Different Functions

A piecewise defined function is literally a function that has been defined in pieces. Most likely you have encountered piecewise defined functions in every day life. Maybe you didn't know what they were called.

For instance...

A tax proposal is being considered in order to simplify income tax filing. The tax a person will pay depends on their amount of income.

Figure 2.2.2. Graph of Income Tax function (pieceswise example)

Annual incomes of less than or equal to \(60\) (thousand dollars) are taxed at a flat rate of \(15 \%\text{.}\)

Annual incomes greater than \(60\) (thousand dollars) are taxed at \(15 \%\) for the first \(60\) then \(35 \%\) for any amount over \(60\) thousand dollars.

What piece of the graph do you use to calculate the income tax? It depends on the income. Of the two line segments, use the left one to calculate the tax for incomes up to \(60\) thousand dollars, and use the right segment to calculate the tax for incomes over \(60\) thousand dollars.

The most difficult part of piecewise functions is the notation. In the following exercise you will use a skill you already know (evaluate a function from a graph), and then transition into how to write and understand piecewise notation.

Subsection 2.2.2 One Function Pieced Together

In the previous exercise you identified the equations of three separate lines and you stated the intervals for which each line was to be used. In fact, you stated the domain for each equation. When two or more equations are put together to form one function it's called a piecewise defined function.

The function has to state what all the formulas are and it also has to let the reader know when to use them. In short, the notation has to let the reader know the equations and their domains.

Next we learn how to set up the notation so that a reader can easily identify each formula and its corresponding domain.

To evaluate a piecewise defined function:

  1. Find the domain each input belongs to, and

  2. use the formula that corresponds to that domain to evaluate the function at the given input value.

In the case where the given input belongs to none of the available domains, we say the function is undefined at that value. The word undefined means the function cannot use the given input so there is no way to return an output value.

Subsection 2.2.3 Graphing a Piecewise Defined Function

We can now use our understanding of piecewise notation to read piecewise functions and graph them. Remember, the domain of a function is the collection of inputs for which the function exists. On a graph, a function is visible only for inputs in its domain and it is not visible for inputs outside its domain.