section 3.3 piece functions
DESCRIPTION
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 afunction. Definition. - PowerPoint PPT PresentationTRANSCRIPT
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Section 3.3
Piece Functions
Section 3.3
Piece Functions
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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.
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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
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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
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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
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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
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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
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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
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Graph ƒ(x) = [x] Graph ƒ(x) = [x]
y
x
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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.
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Practice: Find f(2.75) for the function f(x) = [x].Practice: Find f(2.75) for the function f(x) = [x].
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Practice: Find f(-0.9) for the function f(x) = [x].Practice: Find f(-0.9) for the function f(x) = [x].
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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
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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)}
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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
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Graph ƒ(x) = |x|Graph ƒ(x) = |x|
x if x 0-x if x 0
f(x) = |x| =
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Plot the points (-3, 3), (-2, 2), (0, 0), (1, 1), (3, 3) and connect them to get the following.
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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)}
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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.
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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.
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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.
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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
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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.
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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
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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
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Homework:
pp. 123-125
Homework:
pp. 123-125
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►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]
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►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
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►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|
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►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
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►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]
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►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
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■ 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°.
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■ Cumulative Review38. Find the sine, cosine, and tangent of
2/3.
■ Cumulative Review38. Find the sine, cosine, and tangent of
2/3.
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■ 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.
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■ 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).
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■ 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.