1. MAPPINGS and FUNCTIONS
C3 CORE MATHEMATICS
MAPPINGS and FUNCTIONS
KEY CONCEPTS:
DEFINITION OF A FUNCTION
DOMAIN
RANGE
INVERSE FUNCTION

2. MAPPINGS and FUNCTIONS

3. What is a Function? ?

4. 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.
DOMAIN
The DOMAIN is the set of ALLOWED INPUTS TO A FUNCTION.
RANGE
The RANGE is the set of POSSIBLE OUTPUTS FROM A FUNCTION

5. 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.
WOW!
Only a one-to-one
or a many-to-one
mapping can be called
a function.

6. ONE TO ONE MAPPING MANY TO ONE MAPPING
DOMAIN
RANGE
DOMAIN
RANGE
RANGE
RANGE
DOMAIN
DOMAIN

7. ONE TO MANY MAPPING MANY TO MANY MAPPING2 -2 √8
-√8

8. THESE MAPPINGS ARE NOT FUNCTIONS
ONE TO MANY MAPPING
MANY TO MANY MAPPING 2 -2 √8
-√8
THESE
MAPPINGS
ARE NOT
FUNCTIONS

9. FUNCTIONS (well…almost)
NOT FUNCTIONS
One-one
mapping
Many-one
One-Many
Many-Many
Place the following mappings in the table

10. FUNCTIONS (nearly!)
NOT FUNCTIONS
One-one
mapping
Many-one
One-Many
Many-Many
Place the following mappings in the table

11. BUT THERE’S MORE TO CONSIDER
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.
KNOW
ALL!!
BUT THERE’S MORE
TO CONSIDER

12. 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
The MAPPING becomes a FUNCTION when we
define the DOMAIN
The DOMAIN MAY BE DEFINED- to make a mapping become a function
The set of values in the DOMAIN can also be written in
INTERVAL NOTATION

13. INTERVAL/SET NOTATION
I KNEW THAT
Is a symbol standing for the
SET OF REAL NUMBERS
Means that x “is a member of” the
SET OF REAL NUMBERS

14. Finding the RANGE of a function
The RANGE of a function can be visualised as the projection onto the y axis
The RANGE of a ONE TO ONE FUNCTION will depend on the DOMAIN.
Find the RANGE of the function defined as
The RANGE of the function:
INTERVAL NOTATION

15. The RANGE of a MANY TO ONE FUNCTION will
Need careful consideration.
The set of values in the domain written in
INTERVAL NOTATION is
The set of values in the RANGE written in
INTERVAL NOTATION is
Minimum point with coordinate
(0,-1)
What if you have no graph to LOOK AT?
Then you would need to identify any STATIONARY POINTS of the graph
Take care finding the
range of a Many to one
Function

16. The function is ONE TO ONE on the given DOMAIN so

18. CARE!
CARE!

19. DOMAIN f(x) is EQUAL TO RANGE f -1(x)
Finding the INVERSE FUNCTION x y
DOMAIN
f(x)
RANGE
f-1(x)
For an INVERSE to EXIST the original function MUST BE ONE TO ONE
DOMAIN f(x) is EQUAL TO RANGE f -1(x)
RANGE f(x) is EQUAL TO DOMAIN f -1(x)

20. A function is defined as
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.

21. We see that the graph of the inverse function is
the reflection in the line y=x, of the graph of the function.
VICA VERSA

22. and find the domain and range of
EXAMPLE
The function f is defined as
and
Find an expression for
and find the domain and range of

36. A SPECIAL PAIR OF FUNCTIONS