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Warm-Up
Domain Range
INQ:______________
INT:______________
Is it a function?_______
Is it discrete, continuous?
f(x)=sin(x*10.5/10)
-3 -2 -1 1 2 3
-2
-1
1
2
x
y
f(x)=-x+2
-2 -1 1 2 3 4
-2
-1
1
2
3
4
x
y
Domain Range
INQ:______________
INT:______________
Is it a function?_______
Is it discrete, continuous?
1.
2.
Unit 3Unit 3
PIECEWISE PIECEWISE FUNCTIONSFUNCTIONS
Objectives•I can evaluate piecewise functions.
•I can graph piecewise functions.
Definition:Definition:
Piecewise Function Piecewise Function a function defined by two a function defined by two or more functions over a or more functions over a specified domain.specified domain.
The rule for a piecewise function is different for different parts, or pieces, of the domain. For instance, movie ticketsprices are often different for different ages groups. So the function for movie ticket prices would assign a different value (ticket price) for Each domain interval (age group).
Remember: When using interval notation, square brackets [ ] indicate an included endpoint, and parentheses ( )indicate an excluded endpoint
Evaluating Piecewise Functions:
Evaluating piecewise functions is just like evaluating functions that you are already familiar with.
f(x) = x2 + 1 , x 0x – 1 , x 0
Let’s calculate f(2).
You are being asked to find y when x = 2. Since 2 is 0, you will only substitute into the second part of the function.
f(2) = 2 – 1 = 1
f(x) = x2 + 1 , x 0x – 1 , x 0
Let’s calculate f(-2).
You are being asked to find y when x = -2. Since -2 is 0, you will only substitute into the first part of the function.
f(-2) = (-2)2 + 1 = 5
Your turn:
f(x) = 2x + 1, x 02x + 2, x 0
Evaluate the following:
f(-2) = -3?
f(0) = 2?
f(5) = 12?
f(1) = 4?
One more:
f(x) = 3x - 2, x -2-x , -2 x 1x2 – 7x, x 1
Evaluate the following:
f(-2) = 2?
f(-4) = -14?
f(3) = -12?
f(1) = -6?
Piecewise Function – A function defined in pieces.
3
fx 4 x 0
x 3 x 0x
2
f 3 2 3 3 3
f 6 3 6 14 4
f 5 3 5 14 1
f 2 2 2 3 1
f 0 2 0 33
2x x 3
x
x 1 x 2
xx 4 3 2f
f 0 0 44
f 5 225 5
f 4 4 31
f 3
f 3
f 4
3 21
3 4 1
214 6
2x 5 x 1
g x x 3 1 x 3
3x 1 x 3
g 6
g 2
g 0
g 1
g 3
3 6 11 7
2 1 5 7
0 33
2 13
2 3 5 11
4 x 3
h x 2x 3 3 x 4
4x 7 x 6
h 4
h 3
h 3
h 4
h 5
h 6
4
4
2 3 3 9
2 4 3 11
DNE
4 6 7 17
Graphing Piecewise Functions:
f(x) = x2 + 1 , x 0x – 1 , x 0
Determine the shapes of the graphs.
Parabola and LineDetermine the boundaries of each graph.
Graph the parabola where x is less than zero.
Graph the line where x is greater than or equal to zero.
3x + 2, x -2-x , -2 x 1x2 – 2, x 1
f(x) =
Graphing Piecewise Functions:
Determine the shapes of the graphs.
Line, Line, ParabolaDetermine the boundaries of each graph.
Graphing Piecewise Functions
x 4 x 4
2x
x 3 x
1
1
g x 5 4 x
Domain - ,
Range - , 7
REAL WORLD
The graph shows the monthly fee for Cell Zone. Use it to answer the following questions:
1) What is the monthly fee? 2) How many minutes are included in the monthly
fee? 3) If a customer goes over the minutes included in
the fee, how much will they be charged per minute ($/min)?
4) Write a function for this plan.
100 200 300 400 500 600 700 800
Peak Minutes (minutes)
80 60 40 20
Fee ($)
Lesson Quiz: Part I
1. Graph the function, and evaluate at x = 1 and x = 3.
p(x) =x2 + 2 if x ≤ 2
1 2
x + 3 if x > 2 1 2
Lesson Quiz: Part II
2. Write and graph a piecewise function for the following situation. A house painter charges $12 per hour for the first 40 hours he works, time and a half for the 10 hours after that, and double time for all hours after that. How much does he earn for a 70-hour week?
RECALLRECALL
Domain -
Range -
,
, 4
Domain -
Range -
[-1, 5]
[-5, 3]
Domain -
Range -
(-7, -1), (-1, 7]
[-1, 5), [6, 6]
Domain -
Range -
(-7, 4), [5, 7)
[-7, -5), (-2, 7)
Piecewise Function – Domain and Range
Domain -
Range -
(-6, 7)
[-1, 5)
Domain -
Range -
[-7, 7]
(-4.5, -1], [0, 4)
3 7 x 4
1x 2 4 x 0
2
1
x 4x
0 x 5
5 x 7
g
Domain -
Range -
(-7, 7]
(-4, -2), [-1, 4]
1x 6 x 3
3x 1 3 x 0h x
x 4 0 x 3
x 3 3 x 7
Domain -
Range -
[-6, 7]
[-4, 2], (4, 7)
