Section 3.3

Piece Functions

Section 3.3

Piece Functions

Objectives:1. To define and evaluate piece

functions.2. To graph piece functions and

determine their domains andranges.

3. To introduce continuity of afunction.

Objectives:1. To define and evaluate piece

functions.2. To graph piece functions and

determine their domains andranges.

3. To introduce continuity of afunction.

Piece functions are functions that requires two or more function rules to define them.

Piece functions are functions that requires two or more function rules to define them.

DefinitionDefinitionDefinitionDefinition

EXAMPLE 1 Evaluate f(0) and f(3)

for f(x) = .

EXAMPLE 1 Evaluate f(0) and f(3)

for f(x) = .

-3x + 2 if x 12x if x 1

f(0) = -3(0) + 2 = 2

f(3) = 23 = 8

EXAMPLE 2

Graph f(x) = . Give the

domain and range.

EXAMPLE 2

Graph f(x) = . Give the

domain and range.

-3x + 2 if x 12x if x 1

EXAMPLE 2

Graph f(x) = . Give the

domain and range.

EXAMPLE 2

Graph f(x) = . Give the

domain and range.

D = {real numbers}

R = {y|y -1}

-3x + 2 if x 12x if x 1

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.

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.

DefinitionDefinitionDefinitionDefinition

EXAMPLE 3 Find the set of ordered pairs described by the greatest integers function f(x) = [x] and the domain {-5, -3/2, -3/4, 0, 1/4, 5/2}.

EXAMPLE 3 Find the set of ordered pairs described by the greatest integers function f(x) = [x] and the domain {-5, -3/2, -3/4, 0, 1/4, 5/2}.

f(-5) = [-5] = -5f(-3/2) = [-3/2] = -2

f(0) = [0] = 0f(1/4) = [1/4] = 0

f(5/2) = [5/2] = 2

Graph ƒ(x) = [x] Graph ƒ(x) = [x]

y

x

f(x) = [x] =f(x) = [x] =

......

......

22xx11ifif11

11xx00ifif00

-- 00xx11ifif-1-1

---- 11xx22ifif-2-2

The rule for the greatest integer function can be written as a piece function.The rule for the greatest integer function can be written as a piece function.

Practice: Find f(2.75) for the function f(x) = [x].Practice: Find f(2.75) for the function f(x) = [x].

Practice: Find f(-0.9) 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}.

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| = 11g(-2) = |2(-2) – 3| = |-7| = 7g(0) = |2(0) – 3| = |-3| = 3g(1) = |2(1) – 3| = |-1| = 1g(2) = |2(2) – 3| = |1| = 1g(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}.

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)}

Absolute value function The absolute value function is expressed as {(x, ƒ(x)) | ƒ(x) = |x|}.

Absolute value function The absolute value function is expressed as {(x, ƒ(x)) | ƒ(x) = |x|}.

DefinitionDefinitionDefinitionDefinition

Graph ƒ(x) = |x|Graph ƒ(x) = |x|

x if x 0-x if x 0

f(x) = |x| =

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.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 Graphs1. If x is replaced by x - a, where a

{real numbers}, the graph translates horizontally. If a > 0, thegraph moves a units right, and if a< 0 (represented as x + a), it movesa units left.

Translating Graphs1. If x is replaced by x - a, where a

{real numbers}, the graph translates horizontally. If a > 0, thegraph moves a units right, and if a< 0 (represented as x + a), it movesa units left.

Translating Graphs2. If y, or ƒ(x), is replaced by y - b,

where b {real numbers}, thegraph translates vertically. If b > 0,the graph moves b units up, and ifb < 0 (represented as y + b), itmoves b units down.

Translating Graphs2. If y, or ƒ(x), is replaced by y - b,

where b {real numbers}, thegraph translates vertically. If b > 0,the graph moves b units up, and ifb < 0 (represented as y + b), itmoves b units down.

Translating Graphs3. If g(x) = -ƒ(x), then the functions

ƒ(x) and g(x) are reflections of oneanother across the x-axis.

Translating Graphs3. If g(x) = -ƒ(x), then the functions

ƒ(x) and g(x) are reflections of oneanother across the x-axis.

Practice: Find the correct equation of the translated graph.Practice: Find the correct equation of the translated graph.

1. y = |x – 3| + 12. f(x) = |x + 3| + 13. y = |x + 1| - 34. 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.

Continuous functions have no gaps, jumps, or holes. You can graph a continuous function without lifting your pencil from the paper.

EXAMPLE 6 Graph 2x + 3 if x -2

g(x) = |x| if -1 x 1 .x3 if x 1

EXAMPLE 6 Graph 2x + 3 if x -2

g(x) = |x| if -1 x 1 .x3 if x 1

EXAMPLE 6 Graph 2x + 3 if x -2

g(x) = |x| if -1 x 1 .x3 if x 1

EXAMPLE 6 Graph 2x + 3 if x -2

g(x) = |x| if -1 x 1 .x3 if x 1

Homework:

pp. 123-125

Homework:

pp. 123-125

►A. ExercisesFind the function described by the given rule and the domain {-4, -1/2, 0, 3/4, 2}.

3. h(x) = [x]

►A. ExercisesFind the function described by the given rule and the domain {-4, -1/2, 0, 3/4, 2}.

3. h(x) = [x]

►B. ExercisesWithout 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. ExercisesWithout 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. ExercisesWithout 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. ExercisesWithout 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. ExercisesWithout 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. ExercisesWithout 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. ExercisesGraph. Give the domain and range of each. Classify each as continuous or discountinuous.23. g(x) = [x]

►B. ExercisesGraph. Give the domain and range of each. Classify each as continuous or discountinuous.23. g(x) = [x]

►B. ExercisesGraph. Give the domain and range of each. Classify each as continuous or discountinuous.

29. f(x) =

►B. ExercisesGraph. Give the domain and range of each. Classify each as continuous or discountinuous.

29. f(x) =

4 otherwise4 otherwise

x2 if -2 x 2 x2 if -2 x 2

■ Cumulative Review37. Give the reference angles for the

following angles: 117°, 201°, 295°, -47°.

■ Cumulative Review37. Give the reference angles for the

following angles: 117°, 201°, 295°, -47°.

■ Cumulative Review38. Find the sine, cosine, and tangent of

2/3.

■ Cumulative Review38. Find the sine, cosine, and tangent of

2/3.

■ Cumulative Review39. Classify y = 7(0.85)x as exponential

growth or decay.

■ Cumulative Review39. Classify y = 7(0.85)x as exponential

growth or decay.

■ Cumulative ReviewConsider f(x) = –x² – 4x – 3. 40. Find f(-2) and f(-1/2).

■ Cumulative ReviewConsider f(x) = –x² – 4x – 3. 40. Find f(-2) and f(-1/2).

■ Cumulative ReviewConsider f(x) = –x² – 4x – 3. 41. Find the zeros of the function.

■ Cumulative ReviewConsider f(x) = –x² – 4x – 3. 41. Find the zeros of the function.