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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

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functionsIntro

maps elements from set onto elements in set .

Domain & Range

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FunctionsMappingsFunctions as

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is the Domain of and is the Range of .

Functions

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FunctionsMappingsFunctions as

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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

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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

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functionsIntro

One-to-manyA mapping that maps one element to one or more elements is not a function.

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FunctionsMappingsFunctions as

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functionsIntro

Question...

Which of these is a many-to-one function?

None of the above

FunctionsMappingsFunctions as

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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

   

 

• • • • • •

• • • • • •

One-to-Many

Many-to-One

FunctionsMappingsFunctions as

graphsWorking with

functionsIntro

Working with functions

Next

FunctionsMappingsFunctions as

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functionsIntro

Examples

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FunctionsMappingsFunctions as

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functionsIntro

Your turn

 

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None of the above

FunctionsMappingsFunctions as

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Functions as graphs

 

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FunctionsMappingsFunctions as

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functionsIntro

Functions as graphs

Domain

Range

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FunctionsMappingsFunctions as

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functionsIntro

Functions as graphs

 

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FunctionsMappingsFunctions as

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functionsIntro

Functions as graphs

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FunctionsMappingsFunctions as

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functionsIntro

Functions as graphs

 

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FunctionsMappingsFunctions as

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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.