math 2: unit 5b day 3 how do we solve problems using piecewise functions, step functions, and...
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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!
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.
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)