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Functional dependence between two variables

Functional dependence. Argument ( independent variable ).
Function. Single-valued function. Multiple-valued function.

Two variables x and y are tied by a  functional dependence, if for each value of one of them it is possible to receive by the certain rule one or some values of another.

E x a m p l e . A temperature T of water boiling and atmosphere pressure p are tied by a functional dependence,
because  each value of pressure corresponds to a certain value of the temperature and inversely.
So, if  p = 1 bar, then T = 100C; if  p = 0.5 bar, then T = 81.6C.

A variable, values of which are given, is called an argument or an independent variable; the other variable, values of which are found by the certain rule is called a  function. Usually an argument is marked as  x, and a function is marked as  y . If only one value of function corresponds to each value of argument, this function is called a single-valued function; otherwise, if there are many corresponding values, this function is called a multiple-valued function ( two-valued, three-valued and etc.).

E x a m p l e . A body is thrown upwards;  h  is its height over a ground,  t  is the time, passed from a throwing moment.
h is a single-valued function of  t, but  t is a two-valued function of  h, because the body is on the same
height twice: the first time at an assent, the second time at a fall. The formula

binding variables h and  t ( initial velocity  v0 and an acceleration of a gravity g are constants here ), shows that we have only one value of  h at the given t , and two values of  t at the given  h  ( they are determined by solving the quadratic equation ).


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