Definition of Function

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

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

Independent variable x{x} is also called argument of function.

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

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

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

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

functions

To write that y{y} is a function of x{x} we write that y=f(x),y=ϕ(x),y=F(x){y}={f{{\left({x}\right)}}},{y}=\phi{\left({x}\right)},{y}={F}{\left({x}\right)} etc.

Letters f,ϕ,F,..{f},\phi,{F},.. characterize rule that allows for every value of x{x} find corresponding value of y.{y}. Therefore, if we consider different functions of same argument x{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=y(x){y}={y}{\left({x}\right)}.

If we consider function y=f(x){y}={f{{\left({x}\right)}}} and want to write its particular value when x{x} takes particular value x0{x}_{{0}}, we write f(x0){f{{\left({x}_{{0}}\right)}}}.

For example, if f(x)=11+x2{f{{\left({x}\right)}}}=\frac{{1}}{{{1}+{{x}}^{{2}}}}, g(t)=10t{g{{\left({t}\right)}}}=\frac{{10}}{{t}}, h(u)=1u2{h}{\left({u}\right)}=\sqrt{{{1}-{{u}}^{{2}}}} then

when x=1{x}={1} f(1)=11+12=12{f{{\left({1}\right)}}}=\frac{{1}}{{{1}+{{1}}^{{2}}}}=\frac{{1}}{{2}}; when t=5{t}={5} g(5)=105=2{g{{\left({5}\right)}}}=\frac{{10}}{{5}}={2}; when u=a+b{u}={a}+{b} h(a+b)=1(a+b)2{h}{\left({a}+{b}\right)}=\sqrt{{{1}-{{\left({a}+{b}\right)}}^{{2}}}}.