Type of functions

TYPE OF FUNCTIONS   

ONE – ONE OR INJECTIVE FUNCTION: 

A function f: X →Y is said to be one – one or injective if each element in the domain of a function has a distinct image in the co – domain. Example:  f:RR f(x) = 2x is one – one.

MANY – ONE FUNCTION: 

A function f: X →Y is said to be many one if there are at least two elements in the domain whose images are the same.

EXAMPLE: f: RR given by f(x) = x² is Many – one.

METHODS TO DETERMINE ONE – ONE AND MANY – ONE:

If f(x1) = f (x₂)x₁ = x₂ for every x₁,x₂ in the domain, then ‘f’ is one – one else many – one.

If the function is entirely increasing or decreasing in the domain, then ‘f’ is one – one else many – one.

GRAPHICAL METHOD:

If we draw a line parallel to the x – axis intersect the graph of y  = f(x) at one and only one  point, then f(x) is  one – one and if the line parallel to the x – axis cuts  the graph at more than one different points then f(x) is a many – one function. 

Any continuous function f(x) which has at least one local maxima or local minima is many – one.

All even functions are many one. 

All polynomials of even degree defined on R have at least one local maximum or minima and hence are many one on the domain R. Polynomials of odd degree can be one – one or many – one. 

ONTO FUNCTION OR SURJECTIVE FUNCTION:

A function f: X→Y is said to be a onto function or Surjective function if and only if each element of Y is the image of some element of X i. e. if and only if for every y ∈ Y there exists some x ∈ X such that y = f(x). Thus ‘f’ is onto if f(x) = Y i. e. range = co – domain of function. 

Example: The map f: R →[ –1,1] given by f(x)  = sin x is an onto map.

INTO FUNCTION: A function f: X→Y is said to be an into function if there exists at least one element in the co – domain Y which is not an image of any element in the domain X. Example: The map f: R →R given by f(x) = x₂ is  an into map

                                           Onto                                                               Into

ONE – ONE ONTO MAP OR BIJECTIVE FUNCTION: A function f: X →Y is said to be one – one onto or bijective function if and only if                                                        

• distinct elements of X have distinct images in Y                                  
• each element of Y has at least one pre – image in X.

Example: The map f: X →Y given by f(x)  = 2x is a one – one onto map.