9-2
Piecewise Functions
Objectives
Write and graph piecewise functions.
Use piecewise functions to describe realworld situations.
Vocabulary
piecewise function
Holt Algebra 2
step function
9-2
Piecewise Functions
Notes #1-2
1. A) Create a table for the two pieces
B)Graph the function.
g(x) =
–3x
if x < 2
x+3
if x ≥ 2
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?
Holt Algebra 2
9-2
Piecewise Functions
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, fees for golfing can be different depending on the
time of day. So the function for fees would assign a different
value (green fee) for each domain interval (time).
Holt Algebra 2
9-2
Piecewise Functions
Remember!
When using interval notation, square brackets [ ]
indicate an included endpoint, and parentheses
( ) indicate an excluded endpoint. (Lesson 1-1)
Holt Algebra 2
9-2
Piecewise Functions
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.
Holt Algebra 2
9-2
Piecewise Functions
Example 1
Continued
The domain of the function is
divided into three intervals:
Holt Algebra 2
Weights under 2
[0, 2)
Weights 2 and under 5
[2, 5)
Weights 5 and over
[5, ∞)
9-2
Piecewise Functions
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.
Holt Algebra 2
9-2
Piecewise Functions
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.
Holt Algebra 2
9-2
Piecewise Functions
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.”
Holt Algebra 2
9-2
Piecewise Functions
Example 2A: Evaluating a Piecewise Function
Evaluate each piecewise function for x = –1
and x = 4.
h(x) =
2x + 1
if x ≤ 2
x2 – 4
if x > 2
h(–1) = 2(–1) + 1 = –1
h(4) = 42 – 4 = 12
Holt Algebra 2
Because –1 ≤ 2, use
the rule for x ≤ 2.
Because 4 > 2, use the
rule for x > 2.
9-2
Piecewise Functions
Example 2B
Evaluate each piecewise function for x = –1
and x = 3.
12 if x < –3
f(x) = 15 if –3 ≤ x < 6
20 if x ≥ 6
f(–1) = 15
f(3) = 15
Holt Algebra 2
Because –3 ≤ –1 < 6, use the
rule for –3 ≤ x < 6 .
Because –3 ≤ 3 < 6, use the
rule for –3 ≤ x < 6 .
9-2
Piecewise Functions
You can graph a piecewise function by graphing
each piece of the function.
Holt Algebra 2
9-2
Piecewise Functions
Example 3A: Graphing Piecewise Functions
Graph each function.
g(x) =
1
x+3
4
–2x + 3
if x < 0
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.
Holt Algebra 2
9-2
Piecewise Functions
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.
Holt Algebra 2
●
O
9-2
Piecewise Functions
Example 3B: Graphing Piecewise Functions
Graph each function.
x2 – 3
g(x) =
1
2
x–3
(x – 4)2 – 1
if x < 0
if 0 ≤ x < 4
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.
Holt Algebra 2
9-2
Piecewise Functions
Example 3B Continued
No circle is required at (0, –3) and (4, –1) because
the function is connected at those points.
Holt Algebra 2
9-2
Piecewise Functions
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.
Remember!
The distance formula d = rt can be arranged to
find rates: r = d
t
Holt Algebra 2
.
9-2
Piecewise Functions
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 Jennifer’s
distance versus time. Then write a piecewise
function for the graph.
Holt Algebra 2
9-2
Piecewise Functions
Example 4 Continued
Step 1 Make a table to organize the data. Use
the distance formula to find Jennifer’s rate
for each leg of the race.
Holt Algebra 2
9-2
Piecewise Functions
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 < t ≤ 1.5
She bikes for the next hour.
Running: 1.5 < t ≤ 2
She runs the final half hour.
Holt Algebra 2
9-2
Piecewise Functions
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.
Holt Algebra 2
9-2
Piecewise Functions
Example 4 Continued
Step 4 Write a linear function for each leg.
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
The function rule is d(t) =
t
if 0 ≤ t ≤ 0.5
12t – 5.5 if 0.5 < t ≤ 1.5
6t + 3.5
Holt Algebra 2
if 1.5 < t ≤ 2
9-2
Piecewise Functions
Notes #1
1. A) Create a table for the two pieces
B)Graph the function.
g(x) =
–3x
if x < 2
x+3
if x ≥ 2
The function is composed of two linear
pieces. The domain is divided at x = 2.
Holt Algebra 2
9-2
Piecewise Functions
Notes #1
x
–3x
–4
12
–2
6
0
0
2
–6
4
x+3
●
5
7
O
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.
Holt Algebra 2
9-2
Piecewise Functions
Notes #2
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?
Holt Algebra 2