Concept Byte: Piecewise Functions

For Use With Lesson 2-4

Recall from Lesson 1-6 that | x | , vertical line x vertical line comma  the absolute value of x, is the distance of x from zero. When x ≥ 0 , | x | = x . x greater than or equal to 0 comma vertical line x vertical line equals x .  When x < 0 , | x | = − x . x less than 0 comma vertical line x vertical line equals negative x .  The absolute value function is an example of a piecewise function. A piecewise function has different rules for different parts of its domain.

Example 1

Graph the absolute value function f ( x ) = { x , for  x ≥ 0 − x , for  x < 0 . f , open x close , equals . left brace . table with 2 rows and 2 columns , row1 column 1 , x comma , column 2 for , x greater than or equal to 0 , row2 column 1 , negative x comma , column 2 for , x less than 0 , end table . .

Graph each piece separately.

Example 2

Use a graphing calculator to graph the function f ( x ) = { − 2 x + 3 , if  x < 2 x − 1 , if  x ≥ 2 f , open x close , equals . left brace . table with 2 rows and 1 column , row1 column 1 , negative 2 x plus 3 comma if x less than 2 , row2 column 1 , x minus 1 comma if x greater than or equal to 2 , end table

Define the function f(x). Use the brackets from the math expression templates to enter the piecewise function.

Enter the rules for the two branches. Set f 1 f sub 1 (x) = f(x) and graph. Image Long Description

Step functions are piecewise functions. A step function pairs every number in an interval with a single value. The graph of a step function can look like the steps of a staircase. One step function is the greatest integer function y = [x], where [x] represents the greatest integer less than or equal to x.

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