Example 3

What is the graph of the function f ( x ) = [ x ] ? f open x close equals left bracket x right bracket question mark

Each piece of the graph is a horizontal segment that is missing its right endpoint. The open circle indicates that the right endpoint is not part of the graph.

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Given a piecewise function in one form, you can represent it in each of the other forms, including a table, graph, algebraic function, or verbal statement.

Example 4

Media Postage You want to mail a book that weighs 2.5 lb. The table lists postage for a book weighing up to 3 lb. Define and graph the media-postage function. How much will you pay in postage?

f ( x ) = { 2.23 , for  0 < x ≤ 1 2.58 , for  1 < x ≤ 2 2.93 , for  2 < x ≤ 3 f , open x close , equals . left brace . table with 3 rows and 1 column , row1 column 1 , 2.23 , comma , for , 0 less than x less than or equal to 1 , row2 column 1 , 2.58 , comma , for , 1 less than x less than or equal to 2 , row3 column 1 , 2.93 , comma , for , 2 less than x less than or equal to 3 , end table

Since 2 < 2.5 ≤ 3 , 2 less than 2.5 less than or equal to 3 comma  you will pay $ 2.93 . dollars , 2.93 , .

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Example 5

What piecewise function represents the graph?

Piece 1 When x ≤ − 2 , x less than or equal to negative 2 comma  the rule is f ( x ) = 2 x + 6 . f open x close equals 2 x plus 6 .

Piece 2 When − 2 < x ≤ 0 , negative 2 less than x less than or equal to 0 comma  the rule is f ( x ) = − 2 x − 2 . f open x close equals negative 2 x minus 2 .

Piece 3 When x > 0 , x greater than 0 comma  the rule is f ( x ) = 3 x − 2 . f open x close equals 3 x minus 2 .

f ( x ) = { 2 x + 6 for  x ≤ − 2 − 2 x − 2 , for  − 2 < x ≤ 0 3 x − 2 , for  x > 0 f , open x close , equals . left brace . table with 3 rows and 2 columns , row1 column 1 , 2 x plus 6 , column 2 for , x less than or equal to negative 2 , row2 column 1 , negative 2 x minus 2 comma , column 2 for , minus 2 less than x less than or equal to 0 , row3 column 1 , 3 x minus 2 comma , column 2 for , x greater than 0 , end table

Exercises

1. Graph the sign function. f ( x ) = { − 1 , for  x < 0 0 , for  x = 0 1 , for  x > 0 . f , open x close , equals . left brace . table with 3 rows and 2 columns , row1 column 1 , negative 1 comma , column 2 for , x less than 0 , row2 column 1 , 0 comma , column 2 for , x equals 0 , row3 column 1 , 1 comma , column 2 for , x greater than 0 , end table . .
2. Graph the function f(x) = the least integer greater than x.
3. What piecewise function represents the graph below?

4. Postage In 2008, first-class letter postage was $.42 for up to one ounce and $.17 for each additional ounce up to 3.5 oz. Graph this postage function.

   
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