Definition of Function

Suppose we are given two variables ` x ` and `y` with domain of variation `X` and `Y` respectively.

Suppose that `x` can take any value from `X` without any restrictions. Then variable `y` is called function of variable `x` in domain of its variation `X`, if according to some law or rule we can assign to each value of `x` from `X` one definite value of `y` from `Y`.

Independent variable `x` is also called argument of function.

Set `X` of values that can take independent variable `x` is called domain of function.

Note that sequence is particular case of function where domain is set of all natural numbers.

Set `Y` of values that can take dependent variable `y` 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 `x` and `y`.

In definition we wrote that we assign to each value of `x` one value of `y`. Actually we can to each value of `x` assign more than one value of `y`. 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 `x=a` `y` takes two values).


To write that `y` is a function of `x` we write that `y=f(x),y=phi(x),y=F(x)` etc.

Letters `f,phi,F,..` characterize rule that allows for every value of `x` find corresponding value of `y.` Therefore, if we consider different functions of same argument `x` 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 `y`: `y=y(x)`.

If we consider function `y=f(x)` and want to write its particular value when `x` takes particular value `x_0`, we write `f(x_0)`.

For example, if `f(x)=1/(1+x^2)`, `g(t)=10/t`, `h(u)=sqrt(1-u^2)` then

when `x=1` `f(1)=1/(1+1^2)=1/2`; when `t=5` `g(5)=10/5=2`; when `u=a+b` `h(a+b)=sqrt(1-(a+b)^2)`.