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Section 3.3
Piece Functions
Objectives:
1. To define and evaluate piece
functions.
2. To graph piece functions and
determine their domains and
ranges.
3. To introduce continuity of a
function.
Definition
Piece functions are
functions that requires two or
more function rules to define
them.
EXAMPLE 1 Evaluate f(0) and f(3)
 -3x + 2
for f(x) =  x
2
f(0) = -3(0) + 2 = 2
f(3) = 23 = 8
if x  1
.
if x  1
EXAMPLE 2
 -3x + 2
Graph f(x) =  x
2
domain and range.
if x  1
. Give the
if x  1
EXAMPLE 2
 -3x + 2
Graph f(x) =  x
2
domain and range.
D = {real numbers}
R = {y|y  -1}
if x  1
. Give the
if x  1
Definition
A greatest integer function
is a step function, written as
ƒ(x) = [x], where ƒ(x) is the
greatest integer less than or
equal to x.
EXAMPLE 3 Find the set of ordered
pairs described by the greatest integers
function f(x) = [x] and the domain {-5, -3/2,
-3/ , 0, 1/ , 5/ }.
4
4 2
f(-5) = [-5] = -5
f(-3/2) = [-3/2] = -2
f(0) = [0] = 0
f(1/4) = [1/4] = 0
f(5/2) = [5/2] = 2
Graph ƒ(x) = [x]
y
x
The rule for the greatest integer function
can be written as a piece function.
 ...

 -2
 -1
f(x) = [x] = 
 0
 1

 ...
if - 2  x  - 1
if - 1  x  0
if 0  x  1
if 1  x  2
Practice: Find f(2.75) for the function
f(x) = [x].
Practice: Find f(-0.9) for the function
f(x) = [x].
EXAMPLE 4 Find the function
described by the function rule g(x) =
|2x – 3| for the domain {-4, -2, 0, 1, 2, 4}.
g(-4) = |2(-4) – 3| = |-11| = 11
g(-2) = |2(-2) – 3| = |-7| = 7
g(0) = |2(0) – 3| = |-3| = 3
g(1) = |2(1) – 3| = |-1| = 1
g(2) = |2(2) – 3| = |1| = 1
g(4) = |2(4) – 3| = |5| = 5
EXAMPLE 4 Find the function
described by the function rule g(x) =
|2x – 3| for the domain {-4, -2, 0, 1, 2, 4}.
g = {(-4, 11), (-2, 7), (0, 3), (1, 1), (2, 1),
(4, 5)}
Definition
Absolute value function
The absolute value function is
expressed as {(x, ƒ(x)) | ƒ(x) =
|x|}.
Graph ƒ(x) = |x|
x
f(x) = |x| = 
 -x
if x  0
if x  0
Plot the points (-3, 3), (-2, 2), (0, 0), (1, 1),
(3, 3) and connect them to get the
following.
EXAMPLE 5 Graph f(x) = |x| + 3.
Give the domain and range.
f(x) = |x| + 3
{(-4, 7), (-2, 5), (0, 3), (1, 4), 3, 6)}
Translating Graphs
1. If x is replaced by x - a, where a 
{real numbers}, the graph
translates horizontally. If a > 0, the
graph moves a units right, and if a
< 0 (represented as x + a), it moves
a units left.
Translating Graphs
2. If y, or ƒ(x), is replaced by y - b,
where b {real numbers}, the
graph translates vertically. If b > 0,
the graph moves b units up, and if
b < 0 (represented as y + b), it
moves b units down.
Translating Graphs
3. If g(x) = -ƒ(x), then the functions
ƒ(x) and g(x) are reflections of one
another across the x-axis.
Practice: Find the correct equation
of the translated graph.
1. y = |x – 3| + 1
2. f(x) = |x + 3| + 1
3. y = |x + 1| - 3
4. f(x) = [x – 3] + 1
Continuous functions have no gaps,
jumps, or holes. You can graph a
continuous function without lifting
your pencil from the paper.
EXAMPLE
 2x + 3
g(x) =  |x|
 x3
6 Graph
if x  -2
if -1  x  1 .
if x  1
EXAMPLE
 2x + 3
g(x) =  |x|
 x3
6 Graph
if x  -2
if -1  x  1 .
if x  1
Homework:
pp. 123-125
►A. Exercises
Find the function described by the
given rule and the domain {-4, -1/2, 0,
3/ , 2}.
4
3. h(x) = [x]
►B. Exercises
Without graphing, tell where the
graph of the given equation would
translate from the standard position
for that type of function.
11. f(x) = |x| - 7
►B. Exercises
Without graphing, tell where the
graph of the given equation would
translate from the standard position
for that type of function.
13. y = |x + 4|
►B. Exercises
Without graphing, tell where the
graph of the given equation would
translate from the standard position
for that type of function.
15. y = [x + 1] + 6
►B. Exercises
Graph. Give the domain and range of
each. Classify each as continuous or
discountinuous.
23. g(x) = [x]
►B. Exercises
Graph. Give the domain and range of
each. Classify each as continuous or
discountinuous.
 x2 if -2  x  2
29. f(x) = 
 4 otherwise
■ Cumulative Review
37. Give the reference angles for the
following angles: 117°, 201°, 295°,
-47°.
■ Cumulative Review
38. Find the sine, cosine, and tangent of
2/ .
3
■ Cumulative Review
39. Classify y = 7(0.85)x as exponential
growth or decay.
■ Cumulative Review
Consider f(x) = –x² – 4x – 3.
40. Find f(-2) and f(-1/2).
■ Cumulative Review
Consider f(x) = –x² – 4x – 3.
41. Find the zeros of the function.