Piecewise Functions

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9-2
Piecewise Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
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Warm Up Write the equation of each line in
slope-intercept form.
1. slope of 3 and passes through the point (50,
200)
y 3x 50
2. slope of and passes through the point
(6, 40)
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Objectives
Write and graph piecewise functions. Use
piecewise functions to describe real-world
situations.
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Vocabulary
piecewise function step function
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A piecewise function is a function that is a
combination of one or more functions. The rule
for a piecewise function is different for
different parts, or pieces, of the domain. For
instance, movie ticket prices are often different
for different age groups. So the function for
movie ticket prices would assign a different
value (ticket price) for each domain interval
(age group).
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Example 1 Consumer Application
Create a table and a verbal description to
represent the graph.
Step 1 Create a table
Because the endpoints of each segment of the
graph identify the intervals of the domain, use
the endpoints and points close to them as the
domain values in the table.
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Example 1 Continued
The domain of the function is divided into three
intervals
0, 2)
Weights under 2
2, 5)
Weights 2 and under 5
5, 8)
Weights 5 and over
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Example 1 Continued
Step 2 Write a verbal description.
Mixed nuts cost 8.00 per pound for less than 2
lb, 6.00 per pound for 2 lb or more and less
than 5 lb, and 5.00 per pound for 5 or more
pounds.
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Check It Out! Example 1
Create a table and a verbal description to
represent the graph.
Step 1 Create a table
Because the endpoints of each segment of the
graph identify the intervals of the domain, use
the endpoints and points close to them as the
domain values in the table.
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Check It Out! Example 1 Continued
The domain of the function is divided into three
intervals
Green Fee () Time Range (h)
28 8 A.M. noon
24 noon 4 P.M.
12 4 P.M. 9 P.M.
8, 12)
28
12, 4)
24
4, 9)
12
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Check It Out! Example 1 Continued
Step 2 Write a verbal description.
The green fee is 28 from 8 A.M. up to noon, 24
from noon up to 4 P.M., and 12 from 4 up to 9
P.M.
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A piecewise function that is constant for each
interval of its domain, such as the ticket price
function, is called a step function. You can
describe piecewise functions with a function
rule. The rule for the movie ticket prices from
Example 1 on page 662 is shown.
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Read this as f of x is 5 if x is greater than 0
and less than 13, 9 if x is greater than or equal
to 13 and less than 55, and 6.5 if x is greater
than or equal to 55.
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To evaluate any piecewise function for a specific
input, find the interval of the domain that
contains that input and then use the rule for
that interval.
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Example 2A Evaluating a Piecewise Function
Evaluate each piecewise function for x 1 and x
4.
2x 1 if x 2
h(x)
x2 4 if x gt 2
Because 1 2, use the rule for x 2.
h(1) 2(1) 1 1
Because 4 gt 2, use the rule for x gt 2.
h(4) 42 4 12
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Example 2B Evaluating a Piecewise Function
2x if x 1
g(x)
5x if x gt 1
Because 1 1, use the rule for x 1.
Because 4 gt 1, use the rule for x gt 1.
g(4) 5(4) 20
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Check It Out! Example 2a
Evaluate each piecewise function for x 1 and x
3.
12 if x lt 3
15 if 3 x lt 6
f(x)
20 if x 6
Because 3 1 lt 6, use the rule for 3 x lt 6
.
f(1) 15
Because 3 3 lt 6, use the rule for 3 x lt 6 .
f(3) 15
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Check It Out! Example 2b
Evaluate each piecewise function for x 1 and x
3.
3x2 1 if x lt 0
g(x)
5x 2 if x 0
Because 1 lt 0, use the rule for x lt 0.
g(1) 3(1)2 1 4
Because 3 0, use the rule for x 0.
g(3) 5(3) 2 13
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You can graph a piecewise function by graphing
each piece of the function.
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Example 3A Graphing Piecewise Functions
Graph each function.
x 3 if x lt 0
g(x)
2x 3 if x 0
The function is composed of two linear pieces
that will be represented by two rays. Because the
domain is divided by x 0, evaluate both
branches of the function at x 0.
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Example 3A Continued
For the first branch, the function is 3 when x
0, so plot the point (0, 3) with an open circle
and draw a ray with the slope 0.25 to the left.
For the second branch, the function is 3 when x
0, so plot the point (0, 3) with a solid dot and
draw a ray with the slope of 2 to the right.
?
O
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Example 3B Graphing Piecewise Functions
Graph each function.
x2 3 if x lt 0
x 3 if 0 x lt 4
g(x)
(x 4)2 1 if x 4
The function is composed of one linear piece and
two quadratic pieces. The domain is divided at x
0 and at x 4.
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Example 3B Continued
No circle is required at (0, 3) and (4, 1)
because the function is connected at those
points.
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Check It Out! Example 3a
Graph the function.
4 if x 1
f(x)
2 if x gt 1
The function is composed of two constant pieces
that will be represented by two rays. Because the
domain is divided by x 1, evaluate both
branches of the function at x 1.
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Check It Out! Example 3a Continued
The function is 4 when x 1, so plot the point
(1, 4) with a closed circle and draw a
horizontal ray to the left. The function is 2
when x gt 1, so plot the point (1, 2) with an
open circle and draw a horizontal ray to the
right.
?
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Check It Out! Example 3b
Graph the function.
3x if x lt 2
g(x)
x 3 if x 2
The function is composed of two linear pieces.
The domain is divided at x 2.
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Check It Out! Example 3b Continued
x 3x x 3
4 12
2 6
0 0
2 6 5
4 7
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Add an open circle at (2, 6) and a closed circle
at (2, 5) and so that the graph clearly shows the
function value when x 2.
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Notice that piecewise functions are not
necessarily continuous, meaning that the graph of
the function may have breaks or gaps. To write
the rule for a piecewise function, determine
where the domain is divided and write a separate
rule for each piece. Combine the pieces by using
the correct notation.
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Example 4 Sports Application
Jennifer is completing a 15.5-mile triathlon. She
swims 0.5 mile in 30 minutes, bicycles 12 miles
in 1 hour, and runs 3 miles in 30 minutes. Sketch
a graph of Jennifers distance versus time. Then
write a piecewise function for the graph.
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Example 4 Continued
Step 1 Make a table to organize the data. Use the
distance formula to find Jennifers rate for each
leg of the race.
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Example 4 Continued
Step 2 Because time is the independent variable,
determine the intervals for the function.
Swimming 0 t 0.5
She swims for half an hour.
Biking 0.5 lt t 1.5
She bikes for the next hour.
Running 1.5 lt t 2
She runs the final half hour.
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Example 4 Continued
Step 3 Graph the function.
After 30 minutes, Jennifer has covered 0.5 miles.
On the next leg, she reaches a distance of 12
miles after a total of 1.5 hours. Finally she
completes the 15.5 miles after 2 hours.
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Example 4 Continued
Step 4 Write a linear function for each leg.
Use point-slope form y y1 m(x x1).
Swimming d t
Use m 0.5 and (0, 0).
Biking d 12t 5.5
Use m 12 and (0.5, 0.5).
Use m 6 and (1.5, 12.5).
Running d 6t 3.5
t if 0 t 0.5
12t 5.5 if 0.5 lt t 1.5
The function rule is d(t)
6t 3.5 if 1.5 lt t 2
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Check It Out! Example 4
Shelly earns 8 an hour. She earns 12 an hour
for each hour over 40 that she works. Sketch a
graph of Shellys earnings versus the number of
hours that she works up to 60 hours. Then write a
piecewise function for the graph.
Shellys Earnings Shellys Earnings
Hours worked Pay (/hr)
040 8
gt40 12
Step 1 Make a table to organize the data.
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Check It Out! Example 4 Continued
Step 2 Because the number of hours worked is the
independent variable, determine the intervals for
the function.
She works equal to or less than 40 hours.
0 h 40
She works more than 40 hours.
h gt 40
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Check It Out! Example 4 Continued
Step 3 Graph the function.
Shelly earns 8 per hour for 040. After 40
hours, she earns 12 per hour.
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Check It Out! Example 4 Continued
Step 4 Write a linear function for each leg.
Use a point-slope form y y1 m(x x1).
040 hours 8h
Use m 8.
Hours gt 40 12(h 4) 320
Use m 12 and (40, 320).
8h if 0 h 40
The function rule is f(h)
12(h 40) 320 if h gt 40
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Lesson Quiz Part I
1. Graph the function, and evaluate at x 1 and
x 3.
p(x)
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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?