Introduction
A piecewise function is a function that is defined by two
or more expressions on separate portions of the domain.
Piecewise functions are useful for summarizing a realworld scenario that has multiple conditions. For instance,
the amount withheld from an employee’s paycheck to
fund Social Security benefits depends on that
individual’s income. If a person’s annual income is less
than or equal to a certain limit (which changes each
year), then that person pays a percentage of that annual
income toward Social Security. People earning more
than that limit pay a flat rate or specific amount instead
of a percentage.
6.2.1: Piecewise Functions
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Introduction, continued
In a graph of this situation, the “pieces” of the graph
representing the different tax rates would differ in
appearance. In this lesson, you will learn how to create
and graph piecewise functions to solve everyday
problems, such as determining how much in Social
Security taxes you will pay based on your salary.
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6.2.1: Piecewise Functions
Key Concepts
• A piecewise function can be continuous, with no
break in the graph of the function across a specified
domain, or discontinuous, in which the graph has a
break, hole, or jump.
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6.2.1: Piecewise Functions
Key Concepts, continued
• The notation used to represent the pieces of the same
function over different restricted domains is a brace: {.
When stating the domain of the entire piecewise
function, write the domain for each individual piece
next to the appropriate expression. For example, for
the piecewise function f(x):
ì4.57x; x £ 10
f (x) = í
î3.43x; x > 10
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6.2.1: Piecewise Functions
Key Concepts, continued
• One piece of the function, f(x) = 4.57x, represents the
portion of the function over the restricted domain
x ≤ 10. The other piece, f(x) = 3.43x, represents the
portion of the function for which x > 10.
• On a graph, points of discontinuity or “holes” are
shown as open circles, whereas closed circles are
used to denote defined endpoints.
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6.2.1: Piecewise Functions
Common Errors/Misconceptions
• misinterpreting variables from a word problem
• defining incorrect intervals for the functions
• incorrectly evaluating the function pieces at
specified intervals
• incorrectly noting a closed or open circle on a graph
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6.2.1: Piecewise Functions
Guided Practice
Example 1
Suzanne has a part-time job writing blog posts, which
pays $70 per post if she submits fewer than 5 posts a
week. If she posts more than 5 blogs in a week,
Suzanne is paid 1.5 times the regular amount for each
additional post. Create a piecewise function to represent
Suzanne’s income using this pay scale. How much will
Suzanne be paid if she submits 4 posts in the next
week? How much will she be paid if she submits
6 posts?
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6.2.1: Piecewise Functions
Guided Practice: Example 1, continued
1. Write the equations that make up the
piecewise function.
Use the values in the original problem to create the
equations.
Suzanne’s income is $70 per post, or 70x when the
number of posts submitted is less than or equal to 5.
Therefore, the first equation is f(x) = 70x.
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6.2.1: Piecewise Functions
Guided Practice: Example 1, continued
When Suzanne submits more than 5 posts, her
income is $70 per post for the first 5 posts plus
1.5 times the regular pay for each additional post.
Symbolically, this can be written as
70 • 5 + 1.5[70(x – 5)]. Therefore, the second
equation is f(x) = 350 + 105x – 525, which simplifies
to f(x) = 105x – 175.
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6.2.1: Piecewise Functions
Guided Practice: Example 1, continued
2. Determine the domain for each piece of the
function.
Start with the domain for which the first piece,
f(x) = 70x, is valid.
The pay rate of $70 is only valid when x ≤ 5. Since
Suzanne has to write at least 1 post to be paid, the
domain is also limited to values greater than or equal to
1. Therefore, the domain of f(x) = 70x is 1 ≤ x ≤ 5.
Find the domain for the second piece, f(x) = 105x – 175.
The second piece is valid when Suzanne posts more
than 5 blogs. Therefore, the domain of f(x) = 105x – 175
is x > 5.
6.2.1: Piecewise Functions
10
Guided Practice: Example 1, continued
3. Combine the results of steps 1 and 2 to
write the piecewise function.
When writing a piecewise function, write the equation
of each piece to the right of a brace, with its domain
next to it. Each piece and domain is grouped on a
separate line.
The piecewise function that represents Suzanne’s
ì70x; 1£ x £ 5
weekly income is f (x) = í
.
î105x - 175; 5 < x < ¥
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6.2.1: Piecewise Functions
Guided Practice: Example 1, continued
Note that x is not less than 1, since she cannot submit
less than a whole post, nor can she submit a negative
post. Therefore, in either piece of the function, x is
greater than or equal to 1.
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6.2.1: Piecewise Functions
Guided Practice: Example 1, continued
4. Use the piecewise function to calculate
Suzanne’s income for 4 posts in one week.
If 4 posts are submitted, Suzanne’s income is based
on the first equation of the piecewise function,
f(x) = 70x, because 4 ≤ 5. Substitute 4 for x in the first
piece of the function and then solve.
f(x) = 70x
First piece of the function
f(4) = 70(4)
Substitute 4 for x.
f(4) = 280
Simplify.
Suzanne’s income for submitting 4 posts in one week
is $280.
6.2.1: Piecewise Functions
13
Guided Practice: Example 1, continued
5. Use the piecewise function to calculate
Suzanne’s income for 6 posts.
For 6 posts, Suzanne’s income is based on the
second equation of the piecewise function,
f(x) = 105x – 175, because 6 > 5. Substitute 6 for x in
the second piece of the function and then solve.
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6.2.1: Piecewise Functions
Guided Practice: Example 1, continued
f(x) = 105x – 175
Second piece of the
function
f(6) = 105(6) – 175
Substitute 6 for x.
f(6) = 630 – 175
Simplify.
f(6) = 455
Subtract.
Suzanne’s income for submitting 6 posts in one week
is $455.
✔
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6.2.1: Piecewise Functions
Guided Practice: Example 1, continued
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6.2.1: Piecewise Functions
Guided Practice
Example 3
Use the following
graph to identify the
domain of the
function and where
the graph is
continuous and
discontinuous.
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6.2.1: Piecewise Functions
Guided Practice: Example 3, continued
1. Identify the domain of the piecewise
function.
The domain of the function is the x-values for which
the function exists.
It can be seen from the graph that the function exists
on the intervals (0, 1] and (2, ).
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6.2.1: Piecewise Functions
Guided Practice: Example 3, continued
2. Use the graph to identify where the
function is continuous and discontinuous.
The function is continuous on intervals where the
graph is a single unbroken curve. Therefore, the
function is continuous on the intervals (0, 1], (2, 5],
and (5, ).
The function is discontinuous at the x-values that
are represented by a break or a jump in the graph.
Therefore, this function is discontinuous on
the intervals (– , 0), (1, 2], and at the
point x = 5.
✔
6.2.1: Piecewise Functions
19
Guided Practice: Example 3, continued
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6.2.1: Piecewise Functions