 # 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 $\in$X, f(x) = xPx - 1}. Then the range of f is

Marks:1

{1, 2, 6, 24}

##### Explanation:

$f\left(4\right)=4!=4×3×2×1=24$

• Q2

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

Marks:1

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)

• Q3 Marks:1

d = –a

##### Explanation: • 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 2.

##### Explanation:

f(g(x)) = f(x2 + 7) = 2(x2 + 7) + 3 = 25 2x2 = 8 x = 2  