# Category: Function Concept

## Variable Interval

Recall that variable $$${x}$$$ is defined by the set $$${X}$$$ of those values that it can take (in considered question). This set $$${X}$$$ that contains each value of $$${x}$$$ exactly once is called domain of variation of the variable $$${x}$$$.

## Dependence between Variables

We are not really interested in changing of one variable, it is more interesting to study dependence between two variables when they both change.

Such variables can not take simultaneously any pair of values: if one of them (independent variable) is given certain value, then we can find value of second variable (dependent variable).

## 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 $$${Y}$$$.

## Representing a Function

There are three possible ways to represent a function:

- analytically (by formula);
- numerically (by table of values);
- visually (by graph).

Analytical representation of a function.

This is the most common way to represent a function. We define a formula that contains arithmetic operations on constant values and the variable $$${x}$$$ that we need to perform to obtain the value of $$${y}$$$.

## Inverse of a Function

We already know that a function is a rule that allows finding for every value of $$${x}$$$ the corresponding value of $$${y}$$$. But what if we are given a rule, i.e. a function, and the value of $$${y}$$$ and we want to find the corresponding value of $$${x}$$$?