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 D_{f}.

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 R_{j}.

- R
_{f}= {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 x
_{1}≠ x_{2}and x_{1}, x_{2}∈ A and f(x_{1}) ≠ f(x_{2}) 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 x
_{1}≠ x_{2}, x_{1}, x_{2}∈ A and f(x_{1}) = f(x_{1}), 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, x
_{1}≠ x_{2}≠ x_{3}≠ …, x_{1}, x_{2}, x_{3}, … ∈ A and f(x_{1}) = f(x_{2}) = f(x_{3}) =……..= 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 : R
_{f}= {f(x) | x ∈A}, R_{f}= Co-domain; Rf ⊂ (Co-domain) - One-one function : For x
_{1}≠ x_{2}, x_{1}, x_{2}∈ A, f(x_{1}) ≠ f(x_{2}) - Many-one function : For x
_{1}≠ x_{2}, x_{1}, x_{2}∈ A, f(x_{1}) = f(x_{2}) - Constant function : For x
_{1}≠ x_{2}≠ x_{3}, x_{1}, x_{2}, x_{3}∈ A, f(x_{1}) = f (x_{2}) = f(x_{3}) = f (x) - Equal functions : For functions f and g. Domain same and Range same ⇒ f = g