# 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 ${y}={f{{\left({x}\right)}}},{y}=\phi{\left({x}\right)},{y}={F}{\left({x}\right)}$ 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 ${y}$: ${y}={y}{\left({x}\right)}$.

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

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

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