Piecewise FunctionsLesson

Talking Points Included

Algebra I CCSS

An Absolute Value function is a famous piecewise function

An Absolute Value function is a famous piecewise function

It is made up of two pieces

( ) part of the timef x x x ( ) part of the timef x x x

An Absolute Value function is a famous piecewise function

If you try to merge them … you no longer have a function

It fails the vertical line test!

An Absolute Value function is a famous piecewise function

If you try to merge them … you no longer have a function

( ) if 0f x x x x

( ) if 0f x x x x

Unless, of course, you restrict the domain of each portion.

An Absolute Value function is a famous piecewise function

Piecewise Functions 

Functions are not always made up of a single equation. They can be made up of several pieces (if the domain is restricted properly).  The Absolute Value Function This piecewise function is made up of three pieces.is made up of two pieces.         

Evaluate the function for the following values of x.

1. x = 0

2. x = 1

3. x = 5

4. x = -2

5. x = 2

6. x = 10if 0( )

if 0

x xf x x

x x

2 if 2

( ) 6 if 2

10 if 2 and 6

x x

f x x

x x x

Graph Piecewise Functions 

Graph the following. Make sure you look at the restricted domain. Tell whether or not it is a function. Evaluate for the specific given values. 

1. 2. 3.             Function? Yes or No Function? Yes or No Function? Yes or No 

f( ) = f( ) = f( ) =

f( ) = f( ) = f( ) =

 

Write Piecewise Functions Write a piecewise function for the following graphs. Make sure you write the restricted domain next to each piece.Then evaluate the function for the given values. 1. 2. 3.

There are special piecewise function called

STEP FUNCTIONS

and

GREATEST INTEGER FUNCTIONS.

A step function is a piecewise function consisting only

of horizontal lines, or constants.

The greatest integer function, written as

is a step function that yields the greatest integer that is not larger than x.

( )f x x

Greatest Integer Function  

For all real numbers, x, the greatest integer function returns the largest integer LESS THAN OR EQUAL TO x.

In other words, the greatest integer function rounds DOWN a real number to the nearest integer. You can think of the solution as the “greatest integer” up to that point. 

Evaluate the greatest integer function for the following values of x. 

1. x = 4

 

2. x = 1.2

 

3. x = 6.1

 

4. x = 6.9

 

5. x = 103.5

 

6. x = -1.5

 

7. x = -4.8

 

8. x = -10.2

y x

Extra examples

of Piecewise Functions

You can evaluate a piecewise function

You have to pay particularly close attention to the domain so that you

know which portion of the equation to plug your value into.

See if you can write the equations for the piecewise function g(x), restricting the domain appropriately.Note: there are three pieces

See if you can write the equations for the piecewise function f(x), restricting the domain appropriately.Note: there are three pieces