Function as a Special Kind of Relation
Definition: Let and be two non-empty sets. A relation from to , i.e., a sub-set of , is called a function from to , if
(i) for each there exists such that
(ii) and .
Therefore a non-void subset of is a function from to if each element of appears in some ordered pair in f $ and-no two ordered pairs in have the same first element. If then is called the image of under .
Function as a Correspondence
Definition: Let and be two non-empty sets. Then a function from set to set is a rule or method or correspondence which associates elements of set to elements of set such that:
i) all elements of set are associated to elements in set .
ii) an element of set is associated to a unique element in set .
In other words, a function from a set to a set associates each element of to a unique element of set .
Let be a function such that the set consists of a finite number of elements.Then, be described by listing, the values which it attains at different points of its domain.
Domain, Co-domain and Range of a Function
Let . Then the set is known as the domain of and the set is known as the co-domain of . The set of all images of elements of is known as the range of or image set of under and is denoted by .
Example: Let and . Consider a rule . Under this rule, we obtain and . We observe that each element of is associated to a unique element of . So, given by is a function.
Clearly, domain and range .
Equal Functions : Two functions and are said to be equal if and only if (i) domain of domain of (ii) co-domain of co-domain of , and (iii) for every belonging to their common domain. If two functions and are equal, then we write .
Example: Let and given by and given by . Then, we observe that and have the same domain and co-domain. Also we have, and . Hence, .
Real Functions: functions having domain and co-domain both as subsets of the set $latex R of all real number. Such functions are called real functions or real valued functions.
Definition: A function is called a real valued function, if is a subset of ( set of all real numbers).
If both and both are subsets of , then f is called a real function.
Domain of a Real Function: the domain of the real function is the set of all those real numbers for which the expression for or the formula assumes real values only.
Range of a Real Function: The range of a real function of a real variable is the set of all real values taken by at points in its domain.
Operations on Real Functions
Addition: If and be two real functions. Then, their sum is defined as that function from to which associates each to the number .
Product: If and be two real functions. Then, their product is a function from to and is defined as
Subtraction: If and be two real functions. Then, their difference is defined as
Quotient: If and be two real functions. Then, their quotient is a function from to and is defined as
Multiplication of a Function by a Scalar: Let be a real function and be a scalar (real number). Then the product is a function from to and is defined as
Reciprocal of a Function: If is a real function , then its reciprocal function is a function from to and is defined as