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