Post on 01-Apr-2015
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Introduction to functions
Department of MathematicsUniversity of Leicester
Content
Functions
Introduction
Functions as graphs
Working with functions
Mappings
Introduction
FunctionsMappingsFunctions as
graphsWorking with
functions
Next
The term function was coined by mathematician Gottfried Leibniz in 1673 although Oresme came close to a modern formulation of the concept of a function in the 1300s.
The notation “” was introduced my Clairaut and Euler in 1734.
Many famous mathematicians have devoted time to the study of functions including Bernoulli, Fourier and Weierstraß.
Intro
Mappings
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FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
maps elements from set onto elements in set .
Domain & Range
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FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
is the Domain of and is the Range of .
Functions
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FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
A function is a special type of mapping that maps every element of to one element of .
One-to-oneA mapping that maps one element to one element is called a one-to-one function.
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FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Many-to-oneA mapping that maps more than one element to one element is called a many-to-one function.
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FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
One-to-manyA mapping that maps one element to one or more elements is not a function.
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FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Question...
Which of these is a many-to-one function?
None of the above
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Question...
What is this: ?
One-to-One
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One-to-Many
Many-to-One
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Question...
What is this: ?
One-to-One
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• • • • • •
One-to-Many
Many-to-One
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Working with functions
Next
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Examples
Next
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Your turn
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None of the above
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Functions as graphs
Next
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Functions as graphs
Domain
Range
Next
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Functions as graphs
Next
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Functions as graphs
Next
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Functions as graphs
Next
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
Question...
FunctionsMappingsFunctions as
graphsWorking with
functionsIntro
What kind of mapping is this?
One-to-0neOne-to-many
Many-to-one
Conclusion
FunctionsMappingsFunctions as
graphsWorking with
functions
Next
Intro
You should now be able to:
Find the range and domain of a mapping.
Categorise mappings (one-to-one, etc.)
Do simple computations with functions.
When you feel comfortable move on to composition of functions.