MAPPINGS and FUNCTIONS

C3 CORE MATHEMATICS

KEY CONCEPTS:DEFINITION OF A FUNCTIONDOMAINRANGEINVERSE FUNCTION

MAPPINGS and FUNCTIONS

2 1y x

What is a Function?

?

A function is a special type of mapping such that each member of the domain is mapped to one, and only one, element in the range.DOMAINThe DOMAIN is the set of ALLOWED INPUTS TO A FUNCTION.RANGEThe RANGE is the set of POSSIBLE OUTPUTS FROM A FUNCTION

A function is a special type of mapping such that each member of the domain is mapped to one, and only one, element in the range.Only a one-to-one or a many-to-one

mapping can be called a function.

WOW!

ONE TO ONE MAPPING MANY TO ONE MAPPING

DOMAIN RANGE DOMAIN RANGE

2 1y x 2y x

RAN

GE

DOMAIN DOMAINRA

NGE

ONE TO MANY MAPPING

y x

MANY TO MANY MAPPING• 2• -2• 0• √8• -√8

• 2• -2• 0• √8• -√8

2 2 8x y

ONE TO MANY MAPPING

y x

MANY TO MANY MAPPING• 2• -2• 0• √8• -√8

• 2• -2• 0• √8• -√8

2 2 8x y

THESEMAPPINGSARE NOT

FUNCTIONS

FUNCTIONS (well…almost) NOT FUNCTIONS

One-onemapping

Many-onemapping

One-Manymapping

Many-Manymapping

     

     

3y x 1yx

2 3y x 2 2

14 9x y

y x 4y x3y x 3y x

Place the following mappings in the table

FUNCTIONS (nearly!) NOT FUNCTIONS

One-onemapping

Many-onemapping

One-Manymapping

Many-Manymapping

     

     

3y x 1yx

2 3y x

y x

4y x3y x

3y x

Place the following mappings in the table

2 2

14 9x y

A function is a special type of mapping such that each member of the domain is mapped to one, and only one, element in the range.BUT THERE’S MORE TO CONSIDER

KNOW ALL!!

FOR A FUNCTION TO EXIST:The DOMAIN MUST BE DEFINED- Or values which can NOT be in the DOMAIN MUST BE IDENTIFIED

Consider the MAPPING 1yx

The MAPPING becomes a FUNCTION when we define the DOMAIN 1( ) 0f x x

x

The DOMAIN MAY BE DEFINED- to make a mapping become a function

1y x

( ) 1 1f x x x …-

The set of values in the DOMAIN can also be written in INTERVAL NOTATION 1,

I KNEW THAT

INTERVAL/SET NOTATION

R Is a symbol standing for theSET OF REAL NUMBERS

xRMeans that x “is a member of” theSET OF REAL NUMBERS

Finding the RANGE of a function

The RANGE of a ONE TO ONE FUNCTION will depend on the DOMAIN.

The RANGE of a function can be visualised as the projection onto the y axis

Find the RANGE of the function defined as

2( ) -1 1f x x x

The RANGE of the function:

INTERVAL NOTATION

2( ) -1 > 2f x x x

The set of values in the domain written in INTERVAL NOTATION is 2,

The RANGE of a MANY TO ONE FUNCTION willNeed careful consideration.

The set of values in the RANGE written in INTERVAL NOTATION is

What if you have no graph to LOOK AT?Then you would need to identify any STATIONARY POINTS of the graph

Minimum point with coordinate(0,-1)

Take care finding the range of a Many to one Function

The function is ONE TO ONE on the given DOMAIN so

1(bit bigger than 2)(bigger than 2)-2

f

Function and Domain Range 1 2 , f x x x R

1

2 2 10, 5f x x x

2

3 10 2 , 5f x x x

3

4 2 , 2f x x x

4

5 2 10, 0f x x x

5

6 10 2 , 2f x x x

6

7 2 2, 1f x x x

7

8 2 , 3f x x x

8

9 2 2, 1f x x x

9

10 2 2, 0f x x x

10

,0

20,

,0

, 4

,10

6,

2,

, 9

3,

2,

CARE!

CARE!

Finding the INVERSE FUNCTION 1( )f x

1( )f x

( )f xDOMAINf(x)

RANGEf(x)

DOMAINf-1(x)

RANGEf-1(x)

DOMAIN f(x) is EQUAL TO RANGE f -1(x)RANGE f(x) is EQUAL TO DOMAIN f -1(x)

For an INVERSE to EXIST the original function MUST BE ONE TO ONE

x y

( ) 2 2f x x x 1( )f x

1( )f x

1( )f x( )f x

EXAMPLE A function is defined as

(a)Find the inverse function

(b) Find the domain and Range of

(c) sketch the graphs of and on the same pair of axes.

( ) 2 f x x

1 2( ) 2f x x y xWe see that the graph of the inverse function is the reflection in the line y=x, of the graph of the function.VICA VERSA

The function f is defined as 2

2

3( ) for ,05

xf xx

and ( ) 0 for all values of on the domainf x x

Find an expression for

EXAMPLE

1( )f x and find the domain and range of 1( )f x

( ) xf x e

1( ) lnf x x

( ) xf x e

1( ) lnf x x A SPECIAL PAIR OF FUNCTIONS

1( )f f x x

ln( )xe xln xe x