Function

Function: - It is a mapping in which every element of set A is uniquely associated at the element with set B. The set of A is called Domain of a function and set of B is called co-domain.

Domain: - The set of A is called Domain of function

Co-Domain: - The set of B is called Co-Domain.

Range: - It is a collection of the second values in the ordered pair.

Types of Functions / Properties: -

  • One-to-One (Injective): - One element of the Domain set is connected to one element of the Co-Domain set.
  • Onto (Surjective): - Every element of the Co-domain set have one pre-image.
  • Bijective (1:1 and onto): -
  • Into: - If there exists even a single element in B having no pre-image in a, then f is said to be an into function. Clearly, f is a function from A to B. Now, there exists an element C <- B, having no pre-image in A. so, f is an into function.

  • One-one into: - f x -> y. The function f is called one-one into a function if different elements of x have different unique images of y.
  • Many-one function: - If two or more than two elements of the domain have the same image in B, then f is said to be many – one.
  • Many-one into a function: - The function f is called the many – one function if and only if is both many one and into function.
  • Many-one onto function: - The function f is called many-one onto function if and only if is both many one and onto.
  • Identity function: - The function f is called the identity function if each element of set A has an image on itself.

Compositions of Functions: - 

  • Consider functions, ƒ: A → B and ɡ: B → C. The composition of f with g is a function from A into C defined by (gof)(x) = g[f(x)] and is defined by gof.
  • To find the composition of f and g, first find the image of x under f and then find the image of f(x) under g.
    • If f and g are one-to-one then the function (gof) (gof) is also one-to-one
    • If f and g are onto then the function (gof) (gof) is also onto
    • Composition consistently holds associative property but does not hold commutative property.
  • Example: - 

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