### Function & Domain

Function in R :-
A rule of correspondence  f  is called a real value function if it associates each number  x  of the set  A  to an uniquely determined member  y  (called image of  x ) of  B  , a subset of  r  . it i denoted by  f : A --> B or    y = f(x),  indicating that the real number  y  corresponds to  x  which is an element of  A  under the rule  f  . The set  A  is called domain of the function and the set   f(A) =  (y\  y = f(x) ; where  x  is an element of  A  ) i.e.  the set of image of  A  is called the range of   f  . Also   x  is called an independent variable and y  is called a dependent variable .

The domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. The set of values the function may take is termed the range of the function.
For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases). For a function whose domain is a subset of the real numbers, when the function is represented in an xy Cartesian coordinate system, the domain is represented on the x-axis.

Given a function f:XY, the set X is the domain of f; the set Y is the codomain of f. In the expression f(x), x is the argument and f(x) is the value. One can think of an argument as an input to the function, and the value as the output.
The image (sometimes called the range) of f is the set of all values assumed by f for all possible x; this is the set {f(x) | xX}. The image of f can be the same set as the codomain or it can be a proper subset of it. It is in general smaller than the codomain; it is the whole codomain if and only if f is a surjective function.
A well-defined function must carry every element of its domain to an element of its codomain. For example, the function f defined by
$f(x)=1/x$
has no value for f(0). Thus, the set of all real numbers, R, cannot be its domain. In cases like this, the function is either defined on R−{0} or the "gap is plugged" by explicitly defining f(0). If we extend the definition of f to
$f(x) = \begin{cases} 1/x&x\not=0\\ 0&x=0 \end{cases}$
then f is defined for all real numbers, and its domain is $\mathbb{R}$.
Any function can be restricted to a subset of its domain. The restriction of g : A → B to S, where SA, is written g |S : S → B.

Features of a function   y = f(x)  in R   are as follows :-

1)  y  is uniquely determine for every value of  x  from the domain . The value of  y  corresponding to the value   a  of  x  is denoted by   f(a)  .

2)  For two or more  values of  x , y  can have the same value .

3) The image i.e. the value of   y  for a prescribed value of   x  is obtained by substituting the value of   x  in the equation connecting   x ,  y  to  describe the rule   f  .

The natural domain of a formula is the set of values for which it is defined, typically within the reals but sometimes among the integers or complex numbers. For instance the natural domain of square root is the non-negative reals when considered as a real number function. When considering a natural domain the set of possible values of the function is typically called its range .