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# Defining Functions and Kinds of Functions, Math Assignment Help

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mathematics - Defining Functions and Kinds of Functions, Math

**Function:** A function from A to B is a relation A × B such that no two different ordered pairs of the relation have the same first component and every element of A has an image in B.

It is denoted by f : A → B·

or A ^{1}→ B.

**DOMAIN, RANGE AND CO-DOMAIN OF FUNCTION**

**Domain:** Domain of a function is the set of values of x, when (x, y) belongs to the function.** **

**Range:** Range of a function is the set of value of y, when (x, y) belongs to the function.

**Co-domain:** If (x, y) belong to a function f : A → B then Y is called codomain of the function. Range is a subset of co-domain sometimes the range and co-domain have the same elements.

**Note:**

(i) Each element of the set A must be associated.

(ii) All the elements of the set B neednot have the association

(iii) 'The set of elements of B which are associated is called the 'range' of the function.

(iv) The range will be subset of the co-domain.

**Kinds of functions: **

(1) **The function f:** A → B is called an into function, if there is at least one element of set B which has no pre-image in set A.

(2) **The function f:** A → B is called an onto function if every element of set B has at least one pre-image in set A.

(3) **The function f:** A → B is called one-one if distinct elements have distinct images.

(4) **The function f:** A → B is called many-to-one, if one or more elements of set A there correspond only one element of set B.

**Note:**

(1) One-one is also written as 1 - 1.

(2) An onto function is also called 'surjection'.

(3) A1 - 1 onto function is called a 'bijection'.

**Representation of a function:** A function may be indicated by:

(i) a verbal description

(ii) an arrow diagram

(iii) a tabular form

The table represents a function.

X 0 1 2 3

Y 1 4 5 7

(iv) A formula (called an equation).

The equation y = 2x + 3 represents a function.

(v) Set builder notation, such as f: {(x, y): y = 2x + 3)

**Testing for function:** It can tested whether a given relation is a function or not by using the following tests:

(i) In case of a function, the first set i.e., the domain is fully used up.

(ii) In case of a function, the first members of all the ordered pairs are different.

(iii) In case of a function, each element of the first set has only one image in the second set.

(iv) In case of function, a vertical line will intersect the graph of the function at one point only as shown below:

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**Function:**A function from A to B is a relation A × B such that no two different ordered pairs of the relation have the same first component and every element of A has an image in B.

^{1}→ B.

**DOMAIN, RANGE AND CO-DOMAIN OF FUNCTION**

**Domain:**Domain of a function is the set of values of x, when (x, y) belongs to the function.

**Range:**Range of a function is the set of value of y, when (x, y) belongs to the function.

**Co-domain:**If (x, y) belong to a function f : A → B then Y is called codomain of the function. Range is a subset of co-domain sometimes the range and co-domain have the same elements.

**Note:**

**Kinds of functions:**

**The function f:**A → B is called an into function, if there is at least one element of set B which has no pre-image in set A.

**The function f:**A → B is called an onto function if every element of set B has at least one pre-image in set A.

**The function f:**A → B is called one-one if distinct elements have distinct images.

**The function f:**A → B is called many-to-one, if one or more elements of set A there correspond only one element of set B.

**Note:**

**Representation of a function:**A function may be indicated by:

**Testing for function:**It can tested whether a given relation is a function or not by using the following tests:

**Functions and Kinds of Functions, Math Homework Help, Assignment Help**