4.

7 Piecewise Functions
Essential Question How can you describe a function that is
represented by more than one equation?

Writing Equations for a Function

Work with a partner.
a. Does the graph represent y as a function y
of x? Justify your conclusion. 6

CONSTRUCTING b. What is the value of the function when 4

VIABLE x = 0? How can you tell?

2
ARGUMENTS c. Write an equation that represents the values
To be proficient in math, of the function when x ≤ 0. −6 −4 −2 2 4 6x
you need to justify your
f(x) = , if x ≤ 0 −2
conclusions and
communicate them d. Write an equation that represents the values −4
to others. of the function when x > 0.
−6
f(x) = , if x > 0
e. Combine the results of parts (c) and (d) to write a single description of the function.

, if x ≤ 0
f(x) =
, if x > 0

Writing Equations for a Function

Work with a partner.
a. Does the graph represent y as a function y
of x? Justify your conclusion. 6

b. Describe the values of the function for the 4

following intervals.
2

, if −6 ≤ x < −3
−6 −4 −2 2 4 6x
, if −3 ≤ x < 0
f(x) = −2
, if 0 ≤ x < 3
, if 3 ≤ x < 6 −4

−6

Communicate Your Answer y

6
3. How can you describe a function
that is represented by more than 4
one equation?
2
4. Use two equations to describe
the function represented by −6 −4 −2 2 4 6x
the graph. −2

−4

−6

Section 4.7 Piecewise Functions 217

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 4.7 Lesson What You Will Learn
Evaluate piecewise functions.
Graph and write piecewise functions.

Core Vocabul
Vocabulary
larry Graph and write step functions.
Write absolute value functions.
piecewise function, p. 218
step function, p. 220
Evaluating Piecewise Functions
Previous
absolute value function
vertex form Core Concept
vertex
Piecewise Function
A piecewise function is a function defined by two or more equations. Each
“piece” of the function applies to a different part of its domain. An example is
shown below. y

f(x) = {
x − 2, if x ≤ 0
2x + 1, if x > 0
4

2
●
The expression x − 2 represents
the value of f when x is less than f(x) = 2x + 1, x > 0
or equal to 0. −4 −2 2 4 x
●
The expression 2x + 1 f(x) = x − 2, x ≤ 0
represents the value of f when
x is greater than 0. −4

Evaluating a Piecewise Function

Evaluate the function f above when (a) x = 0 and (b) x = 4.

SOLUTION
a. f(x) = x − 2 Because 0 ≤ 0, use the first equation.
f(0) = 0 − 2 Substitute 0 for x.
f(0) = −2 Simplify.

The value of f is −2 when x = 0.

b. f(x) = 2x + 1 Because 4 > 0, use the second equation.
f(4) = 2(4) + 1 Substitute 4 for x.
f(4) = 9 Simplify.

The value of f is 9 when x = 4.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Evaluate the function.

{
3, if x < −2
f(x) = x + 2, if −2 ≤ x ≤ 5
4x, if x > 5
1. f(−8)
3. f(0)
2. f(−2)
4. f(3)
5. f(5) 6. f(10)

218 Chapter 4 Writing Linear Functions

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 Graphing and Writing Piecewise Functions
Graphing a Piecewise Function

Graph y = { −x − 4, if x < 0
x, if x ≥ 0
. Describe the domain and range.

SOLUTION
y
Step 1 Graph y = −x − 4 for x < 0. Because 4
x is not equal to 0, use an open circle
at (0, −4). 2
Step 2 Graph y = x for x ≥ 0. Because x is y = x, x ≥ 0
greater than or equal to 0, use a closed −4 −2 2 4 x
circle at (0, 0).
−2
The domain is all real numbers.
The range is y > −4. y = −x − 4, x < 0

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Graph the function. Describe the domain and range.

7. y =
−x, {
x + 1, if x ≤ 0
if x > 0
8. y =
x − 2, if x < 0
4x, if x ≥ 0 {
Writing a Piecewise Function
y
Write a piecewise function for the graph. 4

SOLUTION
2
Each “piece” of the function is linear.
Left Piece When x < 0, the graph is the line −4 −2 2 4 x
given by y = x + 3.
−2
Right Piece When x ≥ 0, the graph is the line
given by y = 2x − 1. −4

So, a piecewise function for the graph is

f(x) = { x + 3, if x < 0
2x − 1, if x ≥ 0
.

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Write a piecewise function for the graph.

9. y 10. y
4
3
2
1

−4 −2 2 4 x −4 −2 2 4 x

−2 −2

Section 4.7 Piecewise Functions 219

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 Graphing and Writing Step Functions
A step function is a piecewise function defined by a constant value over each part of
STUDY TIP its domain. The graph of a step function consists of a series of line segments.
The graph of a step y
function looks like 2, if 0 ≤ x < 2
a staircase. 6 3, if 2 ≤ x < 4
4, if 4 ≤ x < 6
4 f(x) =
5, if 6 ≤ x < 8
2
6, if 8 ≤ x < 10
7, if 10 ≤ x < 12
0
0 2 4 6 8 10 12 x

Graphing and Writing a Step Function

You rent a karaoke machine for 5 days. The rental company charges $50 for the first
day and $25 for each additional day. Write and graph a step function that represents
the relationship between the number x of days and the total cost y (in dollars) of
renting the karaoke machine.

SOLUTION
Step 1 Use a table to organize Step 2 Write the step function.

{
the information.
50, if 0 < x ≤ 1
Number Total cost 75, if 1 < x ≤ 2
of days (dollars) f(x) = 100, if 2 < x ≤ 3
0<x≤1 50 125, if 3 < x ≤ 4
150, if 4 < x ≤ 5
1<x≤2 75
2<x≤3 100
3<x≤4 125
4<x≤5 150

Step 3 Graph the step function.

Karaoke Machine Rental

y
175
Total cost (dollars)

150
125
100
75
50
25
0
0 1 2 3 4 5 x
Number of days

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

11. A landscaper rents a wood chipper for 4 days. The rental company charges
$100 for the first day and $50 for each additional day. Write and graph a step
function that represents the relationship between the number x of days and the
total cost y (in dollars) of renting the chipper.

220 Chapter 4 Writing Linear Functions

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 Writing Absolute Value Functions
The absolute value function f(x) = ∣ x ∣ can be written as a piecewise function.

f(x) = {−x,
x,
if x < 0
if x ≥ 0
Similarly, the vertex form of an absolute value function g(x) = a∣ x − h ∣ + k can be
REMEMBER written as a piecewise function.
The vertex form of an
absolute value function is
g(x) = { a[−(x − h)] + k, if x − h < 0
a(x − h) + k, if x − h ≥ 0
g(x) = a∣ x − h ∣ + k, where
a ≠ 0. The vertex of the
graph of g is (h, k). Writing an Absolute Value Function

In holography, light from a laser beam is y

(5, 8) mirror
split into two beams, a reference beam and 8
reference reference
an object beam. Light from the object beam
beam beam
reflects off an object and is recombined 6 object
beam
with the reference beam to form images splitter
on film that can be used to create 4
three-dimensional images. film
2 object plate
a. Write an absolute value function that beam mirror
represents the path of the reference beam. (0, 0)
2 4 6 8 x
b. Write the function in part (a) as a laser
piecewise function.

SOLUTION
a. The vertex of the path of the reference beam is (5, 8). So, the function has the
form g(x) = a∣ x − 5 ∣ + 8. Substitute the coordinates of the point (0, 0) into
the equation and solve for a.
g(x) = a∣ x − 5 ∣ + 8 Vertex form of the function
0 = a∣ 0 − 5 ∣ + 8 Substitute 0 for x and 0 for g(x).
−1.6 = a Solve for a.
So, the function g(x) = −1.6∣ x − 5 ∣ + 8 represents the path of the
reference beam.
STUDY TIP
Recall that the graph of b. Write g(x) = −1.6∣ x − 5 ∣ + 8 as a piecewise function.
an absolute value function
is symmetric about the g(x) = { −1.6[−(x − 5)] + 8, if x − 5 < 0
−1.6(x − 5) + 8, if x − 5 ≥ 0
line x = h. So, it makes
sense that the piecewise Simplify each expression and solve the inequalities.
definition “splits” the So, a piecewise function for g(x) = −1.6∣ x − 5 ∣ + 8 is

{
function at x = 5.
1.6x, if x < 5
g(x) = .
−1.6x + 16, if x ≥ 5

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

12. WHAT IF? The reference beam originates at (3, 0) and reflects off a mirror
at (5, 4).
a. Write an absolute value function that represents the path of the
reference beam.
b. Write the function in part (a) as a piecewise function.

Section 4.7 Piecewise Functions 221

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 4.7 Exercises Dynamic Solutions available at BigIdeasMath.com

Vocabulary and Core Concept Check

1. VOCABULARY Compare piecewise functions and step functions.

2. WRITING Use a graph to explain why you can write the absolute value function y = ∣ x ∣ as
a piecewise function.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–12, evaluate the function. (See Example 1.) In Exercises 15–20, graph the function. Describe the
domain and range. (See Example 2.)
f(x) = {5xx +−3,1, if x < −2
if x ≥ −2 15. y = {
−x, if x < 2
x − 6, if x ≥ 2

g(x) = 3, {
−x + 4, if x ≤ −1
if −1 < x < 2
2x − 5, if x ≥ 2
16. y = { 2x, if x ≤ −3
−2x, if x > −3

3. f(−3) 4. f(−2)
17. y = { −3x − 2, if x ≤ −1
x + 2, if x > −1

5. f(0) 6. f(5) 18. y = { x + 8, if x < 4

4x − 4, if x ≥ 4

7. g(−4)

9. g(0)
8. g(−1)

10. g(1)
{
19. y = x − 1,
1, if x < −3
if −3 ≤ x ≤ 3
−2x + 4, if x > 3

11. g(2) 12. g(5)

13. MODELING WITH MATHEMATICS On a trip, the total

{ 2x + 1,
20. y = −x + 2,
−3,
if x ≤ −1
if −1 < x < 2
if x ≥ 2

distance (in miles) you travel in x hours is represented 21. ERROR ANALYSIS Describe and correct the error in
by the piecewise function
finding f(5) when f(x) = { 2x − 3, if x < 5
.
d(x) = { 55x, if 0 ≤ x ≤ 2
.
x + 8, if x ≥ 5

✗
65x − 20, if 2 < x ≤ 5
How far do you travel in 4 hours? f(5) = 2(5) − 3
=7
14. MODELING WITH MATHEMATICS The total cost
(in dollars) of ordering x custom shirts is represented
22. ERROR ANALYSIS Describe and correct the error in

{
by the piecewise function
x + 6, if x ≤ −2

{
17x + 20, if 0 ≤ x < 25 graphing y = .
1, if x > −2
c(x) = 15.80x + 20, if 25 ≤ x < 50.

✗
14x + 20, if x ≥ 50
y
Determine the total cost of ordering 26 shirts.
4

−5 −3 −1 1x

222 Chapter 4 Writing Linear Functions

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 In Exercises 23–30, write a piecewise function for the 35. MODELING WITH MATHEMATICS The cost to join an
graph. (See Example 3.) intramural sports league is $180 per team and includes
the first five team members. For each additional team
23. y 24. y
member, there is a $30 fee. You plan to have nine
3
2 people on your team. Write and graph a step function
1 that represents the relationship between the number
−1 1 3x −2 2 x p of people on your team and the total cost of joining
−2 the league. (See Example 4.)
−3
36. MODELING WITH MATHEMATICS The rates for a
parking garage are shown. Write and graph a step
25. y 26. y
function that represents the relationship between the
1
−4 −2 2x
number x of hours a car is parked in the garage and
2 4 6x −2 the total cost of parking in the garage for 1 day.
−4

−6

27. y 28. y
2
2

−2 2 x
−2 4x
−2 In Exercises 37–46, write the absolute value function as
−4 a piecewise function.
37. y = ∣ x ∣ + 1 38. y = ∣ x ∣ − 3
29. y 30. y
1
4 39. y = ∣ x − 2 ∣ 40. y = ∣ x + 5 ∣
−4 −2 x
−2 2
41. y = 2∣ x + 3 ∣ 42. y = 4∣ x − 1 ∣
−4
2 4 x 43. y = −5∣ x − 8 ∣ 44. y = −3∣ x + 6 ∣

45. y = −∣ x − 3 ∣ + 2 46. y = 7∣ x + 1 ∣ − 5
In Exercises 31–34, graph the step function. Describe
47. MO
MODELING WITH MATHEMATICS You are sitting

{
the domain and range.
on a boat on a lake. You can get a sunburn from
3, if 0 ≤ x < 2 the sunligh
sunlight that hits you directly and also from the
4, if 2 ≤ x < 4 that reflects off the water. (See Example 5.)
31. f(x) = sunlight th
5, if 4 ≤ x < 6
6, if 6 ≤ x < 8

{
y

−4, if 1 < x ≤ 2
−6, if 2 < x ≤ 3
32. f(x) = 5
−8, if 3 < x ≤ 4
−10, if 4 < x ≤ 5

{
3
9, if 1 < x ≤ 2
6, if 2 < x ≤ 4 1
33. f(x) =
5, if 4 < x ≤ 9 1 3 x
1, if 9 < x ≤ 12

34. f(x) =
0,
1,
{
−2, if −6 ≤
−1, if −5 ≤
if −3 ≤
if −2 ≤
x
x
x
x
<
<
<
<
−5
−3
−2
0
a. Write an absolute value function that represents
the path of the sunlight that reflects off the water.
b. Write the function in part (a) as a piecewise
function.

Section 4.7 Piecewise Functions 223

hsnb_alg1_pe_0407.indd 223 2/4/15 4:04 PM

 48. MODELING WITH MATHEMATICS You are trying to 52. HOW DO YOU SEE IT? The graph shows the total cost
make a hole in one on the miniature golf green. C of making x photocopies at a copy shop.

y Making Photocopies
5
C
40

Total cost (dollars)

3 35
30
25
1
20
1 3 5 7 9 x 15
10
5
a. Write an absolute value function that represents 0
the path of the golf ball. 0 100 200 300 400 500 x
Number of copies
b. Write the function in part (a) as a piecewise
function.
a. Does it cost more money to make 100 photocopies
49. REASONING The piecewise function f consists of two or 101 photocopies? Explain.
linear “pieces.” The graph of f is shown.
b. You have $40 to make photocopies. Can you buy
y more than 500 photocopies? Explain.
4

2 53. USING STRUCTURE The output y of the greatest

integer function is the greatest integer less than or
2 4 x equal to the input value x. This function is written as
f(x) = ⟨x⟩. Graph the function for −4 ≤ x < 4. Is it a
piecewise function? a step function? Explain.
a. What is the value of f(−10)?
b. What is the value of f(8)? 54. THOUGHT PROVOKING Explain why

50. CRITICAL THINKING Describe how the graph of each

piecewise function changes when < is replaced with
y= { 2x − 2, if x ≤ 3
−3, if x ≥ 3
≤ and ≥ is replaced with >. Do the domain and range does not represent a function. How can you redefine y
change? Explain. so that it does represent a function?

a. f(x) = { x + 2, if x < 2
−x − 1, if x ≥ 2 55. MAKING AN ARGUMENT During a 9-hour snowstorm,

{ 1 3 it snows at a rate of 1 inch per hour for the first

—2 x + —2 , if x < 1
b. f(x) = 2 hours, 2 inches per hour for the next 6 hours, and
−x + 3, if x ≥ 1 1 inch per hour for the final hour.

51. USING STRUCTURE Graph a. Write and graph a piecewise function that

{∣ ∣
represents the depth of the snow during the
−x + 2, if x ≤ −2
y= . snowstorm.
x, if x > −2
b. Your friend says 12 inches of snow accumulated
Describe the domain and range.
during the storm. Is your friend correct? Explain.

Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons

Write the sentence as an inequality. Graph the inequality. (Section 2.5)

56. A number r is greater than −12 and 57. A number t is less than or equal
no more than 13. to 4 or no less than 18.

Graph f and h. Describe the transformations from the graph of f to the graph of h. (Section 3.6)
1
58. f(x) = x; h(x) = 4x + 3 59. f(x) = x; h(x) = −x − 8 60. f(x) = x; h(x) = −—2 x + 5

224 Chapter 4 Writing Linear Functions

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