# Definition of the Function

Numerical function with domain D is a correspondence, when for every number x from set D we find by some rule number y, that depends on x.

Variable x is called independent variable (or argument). Number y, that corresponds to number x, is called value of a function f at point x and is denoted by f(x). We denote by letter f given function, i.e. functional dependence between variables x and using record y=f(x).

All values that can take independent variable form domain of the function; it is denoted by D(f).

All values that can take function f(x) (for all x that belong to its domain), form range (or codomain) of the function; it is denoted by E(f).

There are also other ways to define a function, for example, variable y is called function of variable x, if there is defined such dependence that allows for any x unambiguously define value of y.

Consider function y=x^2 , where 1<=x<=3  . This record means that following function is defined: for any x from segment [1,3] value of function at point x is x squared. For example,f(1)=1^2=1, f(2)=2^2=4, f(2.3)=2.3^2=5.29 etc.

Record f(4) in this case doesn't make sense because number 4 doesn't belong to segment [1,3]. Segment [1,3] is domain of this function.

Remember, that y=f(x) is just a notation for the function. We could use other letters as well. For example, v=g(u). Here u is independent variable, v is dependent variable, g is a function. Let's see a couple of other examples: if   then  ,  ,  .