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
.
Example: Reference – From Wikipedia, the free encyclopedia
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
.
Thus,
Also
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.
Standard Real Functions and their Graphs (to be linked)
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
.
for all
Product: If and
be two real functions. Then, their product
is a function from
to
and is defined as
for all
Subtraction: If and
be two real functions. Then, their difference
is defined as
for all
Quotient: If and
be two real functions. Then, their quotient
is a function from
to
and is defined as
for all
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
for all
Reciprocal of a Function: If is a real function , then its reciprocal function
is a function from
to
and is defined as