Transcript
Functions
Mathematics 4
June 20, 2012
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DefinitionsRelations
Relations
A set of ordered pairs (x, y) such that for each x-value, there correspondsat least one y-value.
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DefinitionsFunction
Function
A set of ordered pairs (x, y) such that for each x-value, there correspondsexactly one y-value.
Function
A correspondence from a set X ⊆ R to a set Y ⊆ R where x ∈ X andy ∈ Y , and y is unique for a specific value of x.
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DefinitionsOne-to-one Function
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DefinitionsDomain and Range
Domain
The domain of a function is the set of all possible values of x(independent variable, abscissa) for a given relation or function.
Range
The range of a function is the set of all possible values of y (dependentvariable, ordinate) for a given relation or function.
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ExamplesDefinitions of Functions, Domain and Range
Identify if the following relations are functions, and give the domainand range.
1 y = x2 + 6x+ 4
2 x2 + y2 = 1
3 y =1
x+ 1
4 y =1
x2 + 1
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ExamplesDefinitions of Functions, Domain and Range
Identify if the following relations are functions, and give the domainand range.
1 y = x2 + 6x+ 4
2 x2 + y2 = 1
3 y =1
x+ 1
4 y =1
x2 + 1
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ExamplesDefinitions of Functions, Domain and Range
Identify if the following relations are functions, and give the domainand range.
1 y = x2 + 6x+ 4
2 x2 + y2 = 1
3 y =1
x+ 1
4 y =1
x2 + 1
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ExamplesDefinitions of Functions, Domain and Range
Identify if the following relations are functions, and give the domainand range.
1 y = x2 + 6x+ 4
2 x2 + y2 = 1
3 y =1
x+ 1
4 y =1
x2 + 1
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Homework 5Identify if the following relations are functions, and give the domain and range.
1 y =3
x+ 1
2 y =3x2 + 1
x2 + 2
3 y =√−x2 + 25
4 x2 + y2 = 100
5 y + 3 = (x+ 4)2
6 y =2
|x|
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Function Notation
Given the equation y = 2x2 + 5
Using the set-builder notation and the definition of functions:
f = {(x, y)∣∣y = 2x2 + 5}
From this notation we can use the shorthand:
f(x) = 2x2 + 5
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Function Notation
Given the equation y = 2x2 + 5
Using the set-builder notation and the definition of functions:
f = {(x, y)∣∣y = 2x2 + 5}
From this notation we can use the shorthand:
f(x) = 2x2 + 5
Mathematics 4 () Functions June 20, 2012 8 / 1
Function Notation
Given the equation y = 2x2 + 5
Using the set-builder notation and the definition of functions:
f = {(x, y)∣∣y = 2x2 + 5}
From this notation we can use the shorthand:
f(x) = 2x2 + 5
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DefinitionsGraphs of Functions
The graph of a function
The graph of a function is the set of ALL POINTS in R2 for which(x, y) ∈ a given function.
Vertical Line Test
The graph of a function can be intersected by a vertical line in at mostone point.
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DefinitionsGraphs of Functions
The graph of a function
The graph of a function is the set of ALL POINTS in R2 for which(x, y) ∈ a given function.
Vertical Line Test
The graph of a function can be intersected by a vertical line in at mostone point.
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Example:Square root functions
Find the graph of the function f = {(x, y)∣∣y =
√4− x}:
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Example:Square root functions
Find the graph of the function f = {(x, y)∣∣y =
√x− 1}:
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Example:Absolute value functions
Find the graph of the function f = {(x, y) |y = |x− 3|}:
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Homework 6Sketch the graph and determine domain and range for each function below.
1 f = {(x, y) | y =√16− x2}
2 g = {(x, y) | y = (x− 1)3}
3 h =
{(x, y) | y =
x2 − 4x+ 3
x− 1
}
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Evaluating Functions
Evaluating Functions
Assign values to the independent variable and simplifying.
Evaluate the following:
1f(x+ h)− f(x)
hif f(x) = 3x2 − 2x+ 4
2 f(−x) if f(x) = 3x4 − 2x2 + 7
3 g(−x) if g(x) = 3x5 − 4x3 − 9x
4 f(x2 − 1) if f(x) =1 + x
2x− 1
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Evaluating Functions
Evaluating Functions
Assign values to the independent variable and simplifying.
Evaluate the following:
1f(x+ h)− f(x)
hif f(x) = 3x2 − 2x+ 4
2 f(−x) if f(x) = 3x4 − 2x2 + 7
3 g(−x) if g(x) = 3x5 − 4x3 − 9x
4 f(x2 − 1) if f(x) =1 + x
2x− 1
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Evaluating Functions
Evaluating Functions
Assign values to the independent variable and simplifying.
Evaluate the following:
1f(x+ h)− f(x)
hif f(x) = 3x2 − 2x+ 4
2 f(−x) if f(x) = 3x4 − 2x2 + 7
3 g(−x) if g(x) = 3x5 − 4x3 − 9x
4 f(x2 − 1) if f(x) =1 + x
2x− 1
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Evaluating Functions
Evaluating Functions
Assign values to the independent variable and simplifying.
Evaluate the following:
1f(x+ h)− f(x)
hif f(x) = 3x2 − 2x+ 4
2 f(−x) if f(x) = 3x4 − 2x2 + 7
3 g(−x) if g(x) = 3x5 − 4x3 − 9x
4 f(x2 − 1) if f(x) =1 + x
2x− 1
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Operations on Functions
The following notations are indicate an operation between two functions:
(f + g)(x) = f(x) + g(x)(f − g)(x) = f(x)− g(x)
(f · g)(x) = f(x) · g(x)(f
g
)(x) =
f(x)
g(x)
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Operations on Functions
Determine the result of the following function operations:
1 (f + g)(x) if f(x) =x+ 3
x+ 2and g(x) = x− 2
2
(f
g
)(x) if f(x) =
√x3 − x2 − 5x− 3 and g(x) =
√x− 3
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Operations on Functions
Determine the result of the following function operations:
1 (f + g)(x) if f(x) =x+ 3
x+ 2and g(x) = x− 2
2
(f
g
)(x) if f(x) =
√x3 − x2 − 5x− 3 and g(x) =
√x− 3
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Composition of FunctionsDefinition
Composition of Functions
Evaluating a function f(x) with another function g(x).
f (g(x)) = (f ◦ g)(x)
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Composition of Functions
Evaluate the following composite functions:
1 f(x) =x+ 1
x− 1, g(x) =
1
x, find (f ◦ g) and (g ◦ f)
2 f(x) =√x2 − 1, g(x) =
√x− 1, find (f ◦ g)(x) and (g ◦ f)
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Composition of Functions
Evaluate the following composite functions:
1 f(x) =x+ 1
x− 1, g(x) =
1
x, find (f ◦ g) and (g ◦ f)
2 f(x) =√x2 − 1, g(x) =
√x− 1, find (f ◦ g)(x) and (g ◦ f)
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Odd and Even FunctionsDefinitions
Even Function
A function f is even if f(−x) = f(x).
Example
1 f(x) = 3x6 − 2x4 + 4x2 + 2
2 g(x) = |x|+ 2
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Odd and Even FunctionsDefinitions
Even Function
A function f is even if f(−x) = f(x).
Example
1 f(x) = 3x6 − 2x4 + 4x2 + 2
2 g(x) = |x|+ 2
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Odd and Even FunctionsDefinitions
Odd Function
A function f is odd if f(−x) = −f(x).
Example
1 f(x) = 2x5 − 4x3 + 5x
2 g(x) =1
x
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Odd and Even FunctionsDefinitions
Odd Function
A function f is odd if f(−x) = −f(x).
Example
1 f(x) = 2x5 − 4x3 + 5x
2 g(x) =1
x
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Odd and Even FunctionsDetermine if the following functions are odd, even, or neither
(1) (2)
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Odd and Even FunctionsDetermine if the following functions are odd, even, or neither
(3) (4)
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Odd and Even FunctionsDetermine if the following functions are odd, even, or neither
(5) (6)
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Odd and Even FunctionsDetermine if the following functions are odd, even, or neither
(7) (8)
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Odd and Even FunctionsSymmetry properties
Even functions
The graph of even functions are symmetric with respect to the y-axis.
Odd functions
The graph of odd functions are symmetric with respect to the origin.
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Odd and Even FunctionsSymmetry properties
Even functions
The graph of even functions are symmetric with respect to the y-axis.
Odd functions
The graph of odd functions are symmetric with respect to the origin.
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Homework 7Determine if the function is odd/even/neither, then find the domain, range, and thegraph of the function.
1 f(x) =x
x2 − 4
2 (f + g)(x) if f(x) = x2 + 1 and g(x) = |x|
3 (f − g)(x) if f(x) = |x|+ 1 and g(x) = x2 − 2
4 (f · g) if f(x) = x and g(x) = x3
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