MAPPINGS and FUNCTIONS

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Presentation transcript:

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

MAPPINGS and FUNCTIONS

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. DOMAIN The DOMAIN is the set of ALLOWED INPUTS TO A FUNCTION. RANGE The 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. WOW! Only a one-to-one or a many-to-one mapping can be called a function.

ONE TO ONE MAPPING MANY TO ONE MAPPING DOMAIN RANGE DOMAIN RANGE RANGE RANGE DOMAIN DOMAIN

ONE TO MANY MAPPING MANY TO MANY MAPPING 2 -2 √8 -√8

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

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

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

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

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

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

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

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

The function is ONE TO ONE on the given DOMAIN so

CARE! CARE!

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)

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.

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

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

A SPECIAL PAIR OF FUNCTIONS