Preparation For Pre-calculus Functions

Introduction to preparation for pre-calculus functions

Theorems  on Pre calculus Functions

Topics: 1)Definitions

2)Theorem on One-One mapping

3)Theorem on Onto mapping

Function :  A relation f , which associates to each element of a set A, a unique element of a set B, is called a function of A to B. This is written as

f  : A ―>B  ( read as f is mapping from A to B or f maps A to B)

The set A is called the domain of the function f and B is called the co- domain of  f .

 

Preparation for pre-calculus functions- definitions

 

One –One Mapping : a mapping f ;A `->` B is called a one- one  function if distinct elements of A have distinct images in B

For all a₁, a₂ є A, f(a₁) = f(a₂) →a₁ =a₂

Onto function : The mapping f :A `->`B is called onto function if every element of B occurs as the image of at least one element of A.

F:A `->` B id onto , if f(A )= B i.e., range of f = co-domain of f

Bijection: If the function f is both one –one and onto then it is called a Bijection function.

 

Preparation for pre-calculus functions – Theorem

 

Preparation for pre-calculus functions  – learning the proofs of some theorems.

Theorem : If f :A ->B and g: B->C are one- one functions then the mapping gof :A->C is one -one.

Proof :  f :A->B ,and g: B->C are one –one

Let a₁,a₂ belongs to A then f(a₁),f(a₂) belongs to B and g(f(a₁)),g(f(a₂)) belongs to C

Now (gof)a₁= (gof)a₂

`=>` g(f(a₁))= g(f(a₂)) `=>` f(a₁) =f(a₂)

 `=>` a₁  =a₂  ( since f  one-one )

Hence gof: A->C is a one –one function.

The converse of the above theorem is not true.

If f: A->B ,g: B->C and gof is one –one, then both f and g need not be one –one

Theorem :If f : A->Band g: B->C are two onto functions then the mapping gof : A->C is Onto.

Proof:        f:A->Band g:B->C are onto

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