Types of Functions

•    Let A and B be two non empty sets. Then a function ‘f’ from set A to set B is a rule or method that associates the elements of set A to the elements of set B such that:

  • All the elements of set A are associated with the elements of set B.
  • An element of set A is associated with a unique element in set B.

•    If the images of the distinct elements of A are distinct, then the function f is known as one-one function.
•    If the images of the distinct elements of A are the same, then the function f is known as many-one function.
•    If for every y belongs to B, there exists an element x in A such that f(x) = y, then f is known as onto function.
•    A function f from A to B is called into iff there exists at least one element in B, which is not the image of any element of A.
•    A function f : A → B is bijective or one - one onto if it is both one - one and onto.
•    A function is said to be surjective function if it is onto function.
•    The composition of functions f : A → B and g : B → C is the function
gof : A → C given by gof(x) = g(f(x)).

•    If f : A → B and g : B  →  C are one-one, then gof : A  → C is also one-one.

•    If f : A → B and g : B  →  C are onto, then gof : A  → C is also onto.

•    If f : A  →  B;  g : B →  C, are two functions such that gof : A → C is defined and one-one, then both f and g are not necessarily one-one.

•    If f : A  →  B is a function and IA, IB are identity functions on A, B respectively, then
(i) foIA = f    (ii) IBof  = f

•    
•    A function f : X → Y is defined to be invertible, if there exists a function
g : Y → X such that gof = Ix and fog = Iy.
 
•    Inverse of a bijective function is unique.

•    The inverse of a bijective function f is also bijective and  (f–1)–1 = f.

•    If f : A →  B is bijective, then (i) f–1of = IA   (ii) fof–1 = IB

•    Let A be a non-empty set and f : A → A, g : A → A be two functions such that
gof = IA = fog , then f and g are both bijectives and g = f–1.
 

Keywords: Function, Types of functions, Composition of functions, One-one function, Many-one function, Onto function, Into function, One-one onto function, Invertible functions, Properties of Invertible Functions, Properties of composition of functions

 

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  • Q1

    Let X = {1, 2, 3, 4}. A function is defined from X to N as R = {(x, f(x)): x X, f(x) = xPx - 1}. Then the range of f is

    Marks:1
    Answer:

    {1, 2, 6, 24}

    Explanation:

    fx=xPX-1=X!X-X+1!=X!     [ nPr=n!n-r!]

     f(1) = 1! = 1,  f(2) = 2! = 2, f3=3!=3×2×1=6

         f4=4!=4×3×2×1=24

     Range of f = {1, 2, 6, 24}

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  • Q2

    If f(x) = ax + b and g(x) = cx + d, then f[g(x)] – g[f(x)] is equivalent to

    Marks:1
    Answer:

    f(d) – g(b).

    Explanation:

    f(g(x)–g(f(x)) = a(cx + d) + b – {c(ax + b) + d}

    = ad + b – bc – d = (ad + b) – (bc + d)

    = f(d) – g(b)

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  • Q3

    Marks:1
    Answer:

    d = –a

    Explanation:

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  • Q4

    If f : R R and g : R R defined by f(x) = 2x + 3 and g(x) = x2 + 7, then the value of x for which f(g(x)) = 25 will be

    Marks:1
    Answer:

    2.

    Explanation:

    f(g(x)) = f(x2 + 7) = 2(x2 + 7) + 3 = 25

     2x2 = 8  x = 2

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  • Q5

    Marks:1
    Answer:

    f-1(x) = f(x).

    Explanation:
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