Definition of Function
Suppose we are given two variables and with domain of variation and respectively.
Suppose that can take any value from without any restrictions. Then variable is called function of variable in domain of its variation , if according to some law or rule we can assign to each value of from one definite value of from .
Independent variable is also called argument of function.
Set of values that can take independent variable is called domain of function.
Note that sequence is particular case of function where domain is set of all natural numbers.
Set of values that can take dependent variable is called range of function.
When we define a function we don't explictly state range of function, because it is automatically defined from domain of function and law of correspondence between and .
In definition we wrote that we assign to each value of one value of . Actually we can to each value of assign more than one value of . Such function is called multi-valued. In calculus we will consider only one-valued functions and even don't treat multi-valued function as function (see figure: to the right is not a function because when takes two values).
To write that is a function of we write that etc.
Letters characterize rule that allows for every value of find corresponding value of Therefore, if we consider different functions of same argument then we should use different letters to denote them.
Of course we can use any letter to write functional dependence. Often we can even repeat letter : .
If we consider function and want to write its particular value when takes particular value , we write .
For example, if , , then
when ; when ; when .