Math 2: Unit 5BDay 3

How do we solve problems using piecewise functions, step functions,

and greatest integer functions?

2

Today’s Brainteaser

Can you figure out the word/phrase represented by the puzzle?

Square Roots Polygon

Gone

Piecewise Functions

A piecewise function is a function represented by more than a single equation. The domain indicates when to use each equation.

1,452

1,23)(

xifx

xifxxf

x

y

Evaluate the function:

1,452

1,23)(

xifx

xifxxf

1. (5)f 2. (1)f

4. ( 2)f 3. (3)f

To evaluate a piecewise function, look at the domain!

Evaluate the function: 8 3, 0

( )2 1, 0

x if xg x

x if x

1. g(0) 2. g( 4)

Points of discontinuity

X-values where there are gaps in the graphEx:

x

y

1,452

1,23)(

xifx

xifxxf

*because there is a gap in the graph at x = 1, this is a point of discontinuity.

Graphing Piecewise Functions

3, 01. ( )

2 , 0

x if xf x

x if x

x yx y

3, 0y x if x 2 , 0y x if x

Domain:

Range:

Intercepts:

Points of discontinuity:

Graphing Piecewise Functions

4, 22. ( )

3 , 2

x if xf x

x if x

x yx y

4, 2y x if x 3 , 2y x if x

Domain:

Range:

Intercepts:

Points of discontinuity:

Graphing Piecewise Functions

1, 0

3. ( ) 22 3, 0

x if xf x

x if x

x yx y

1, 0

2y x if x 2 3, 0y x if x

Domain:

Range:

Intercepts:

Points of discontinuity:

Graphing Piecewise Functions

, 54. ( ) 2

, 55

x if xf x

x if x

x yx y

, 5y x if x 2

, 55

y x if x

Domain:

Range:

Intercepts:

Points of discontinuity:

Writing an absolute value function as a Writing an absolute value function as a piecewise functionpiecewise function

Graph the absolute value functionFind linear equations that represent each

piece of the graphCreate a domain restriction based on the

vertexWrite the function as a piecewise function

Writing an absolute value function as a Writing an absolute value function as a piecewise functionpiecewise function

1. ( ) 2 4 3f x x

Vertex:

Open up or down?

Left side of vertex: Right side of vertex:

Writing an absolute value function as a Writing an absolute value function as a piecewise functionpiecewise function

2. ( ) 3 2 1f x x

Vertex:

Open up or down?

Left side of vertex: Right side of vertex:

Writing an absolute value function as a Writing an absolute value function as a piecewise functionpiecewise function

3. ( ) 2 4f x x Vertex:

Open up or down?

Left side of vertex: Right side of vertex:

GA Notetaking Guide Page 52 – 53 # 1 – 7 all, #12 – 14 all

Step Function

A piecewise function that is defined by a constant value over each part of its domain.

Its graph resembles a set of stairs

Graph the following step functions

2, 0 1 1, 4 3

3, 1 2 2, 3 21. ( ) 2. ( )

4, 2 3 3, 2 1

5, 3 4 4, 1 0

if x if x

if x if xf x f x

if x if x

if x if x

Graph the following step functions

1, 3 2 3, 4 2

3, 2 1 2, 2 03. ( ) 4. ( )

4, 1 2 0, 0 3

6, 2 3 3, 3 6

if x if x

if x if xf x f x

if x if x

if x if x

Greatest integer function: y = [x]

The greatest integer function, y = [x] is defined as follows: [x] is the greatest integer that is less than or equal to x.

Reflection of the greatest integer function

Graph using transformations

y=[x+3] y = [x-1]

Graph using transformations

y=[x]+2 y = [x-2]+1

Real world applications of step functions

In 2005 the cost C (in dollars) to send US postal service express mail up to 3 pounds depended on the weight w (in ounces) according to the function below. What is the cost to send a package weighing: 13 ounces? 3 ounces? 20 ounces? 5 ounces?

5.50, 0 3

6.00, 3 8 ( )

6.50, 8 12

7.00, 12 24

if w

if wC w

if w

if w

REMEMBER: Test is ___________!!!! Do the REVIEW!!!HW: Page 52 (# 5 – 11) and Page 53 (# 5 – 10, 23 – 26)