Post on 10-Jan-2016
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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.
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
NG
E
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 1
yx
2 3y x 2 2
14 9
x 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 1
yx
2 3y x
y x
4y x3y x
3y x
Place the following mappings in the table
2 2
14 9
x 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 1
yx
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)-2f
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 ,0
5
xf x
x
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