# GSEB Class 11 Statistics Notes Chapter 8 Function

This GSEB Class 11 Commerce Statistics Notes Chapter 8 Function covers all the important topics and concepts as mentioned in the chapter.

## Function Class 11 GSEB Notes

Definition of a Function:
If A and B are any two non-empty sets and each element of set A is related with one and only one element of set B by some rule or correspondence f, then f is called a function from A to B. Symbolically, it is denoted by f: A → B.

• The rule or correspondence can also be denoted by g, h, k, etc.
• If the elements of set A are x and that of set B are y, then y is called an image of x. In usual notation it is denoted by y =f(x) or y = y (x) or y = h(x) or y = k(x).

Domain, Co-domain and Range of a Function:
If f: A → B, then

The domain of a function:
Set A of elements x is called domain of a function f It is denoted by Df.

Co-domain of a function:
Set B of elements y is called the co-domain of a function f.

Range of a function:
A set of images or functional values (y or f(x)) of all the elements (x) of set A is called range of the function / It is denoted by Rj.

• Rf = {f{x) | x ∈ A}
• Range is co-domain itself or a subset of co-domain.
• In co-domain there can be such elements which are not the images of a element of domain of a function.

Notations of Function:

• f: A → B, f is a function from set A to set B. A = Domain, B = Co-domain
• y : P →S, g is a function from set P to set S. P = Domain, S = Co-domain
• k: X → Y, k is a function from set X to set Y. X = Domain, Y = Co-domain
• h : T →U, h is a function from set T to set U. T = Domain, U = Co-domain Also, F(x), Φ(x) etc. Types of Function:

• One-one Function: Suppose f: A B. If for any two different elements of set A (domain), their images are different in set B (Co-domain), then function f is called one-one function, i.e., for function f: A → B, if x1 ≠ x2 and x1, x2 ∈ A and f(x1) ≠ f(x2) then function / is called one-one function.
• Many-one Function: Suppose, f: A →B. If for any two different elements of set A (domain), their images are same in set B (co-domain), then function f is called many- one function, i.e., for function f: A→ B if x1 ≠ x2, x1, x2 ∈ A and f(x1) = f(x1), then function f is called many-one function.
• Constant Function: Suppose, f: A →B. If for each element of set A (domain), the image is same in the set B (co-domain), then function f is called constant function, i.e., for f :A → B, x1 ≠ x2 ≠ x3 ≠ …, x1, x2, x3, … ∈ A and f(x1) = f(x2) = f(x3) =……..= f(x), then function f is called constant functions.

Equal Functions:
Suppose f and g are two different functions. If these two functions satisfy the following conditions, than they are said to be equal functions :

• Domain of functions f and g should be same, i.e., both functions should be defined on same domain.
• For each element x of domain A, the images of function f and function g, should be same in co-domain, i.e., f(x) = g (x).
• In symbol, the equal functions are denoted by f = g
• If f: A→B and y: A→C, then for each x ∈ A, f(x) = g (x) then f = g.

Real Function:
A function for which domain and co-domain are set of real number or subset of set R, then such function is called real function.

Points to be npted for Function:

• Set: A group of welldefined things.
• x ∈ A : x belongs to A
• N = Set of natural numbers = {1, 2, 3, 4, …}
• Z = Set of integers = {…, -3, -2, -1, 0, 1, 2, 3,…}
• R = Set of real numbers
• Empty set = { },
• Non-empty set = {1, 2, 3, 4,5}
• Function = Rule, Relation, Correspondence
• Notations for function: f, g, h, k, etc.
:f(x), F(x), Φ(x), g(x), h (x), k (x)
• Function defined: For non-empty sets
• Meaning of f: A → B : Function f from A to B
• Set A: Domain
• Set B : Co-domain = Set of images of elements of A
•  Range : Rf = {f(x) | x ∈A}, Rf = Co-domain; Rf ⊂ (Co-domain)
• One-one function : For x1 ≠ x2, x1, x2 ∈ A, f(x1) ≠ f(x2)
• Many-one function : For x1 ≠ x2, x1, x2 ∈ A, f(x1) = f(x2)
• Constant function : For x1 ≠ x2 ≠ x3, x1, x2, x3 ∈ A, f(x1) = f (x2) = f(x3) = f (x)
• Equal functions : For functions f and g. Domain same and Range same ⇒ f = g