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Basic notions and properties of functions

Function. Domain and codomain of a function.
Rule (law) of correspondence. Monotone function.
Bounded and unbounded function. Continuous and
discontinuous function. Even and odd function.
Periodic function. Period of a function.
Zeros (roots) of a function. Asymptote.

Domain and codomain of function. In elementary mathematics we study functions only in a set of real numbers R. This means that an argument of a function can adopt only those real values, at which a function is defined, i.e. it also adopts only real values. A set  X  of all admissible real values of an argument  x, at which a function y = f ( x ) is defined, is called a domain of a function. A set  Y  of all real values  y, that a function adopts, is called a codomain of a function. Now we can formulate a definition of a function more exactly: such a rule (law) of a correspondence between a set X and a set Y, that for each element of a set X one and only one element of a set Y can be found, is called a function. From this definition it follows, that a function is given if :
  - the domain of a function  X  is given;
  - the codomain of a function Y is given;
  - the correspondence rule ( law ), is known.
A correspondence rule must be such, that for each value of an argument only one value of a function can be found. This requirement of a single-valued function is obligatory.

Monotone function. If  for any two values of an argument  x1 and x2 from the condition  x2 > x1  it follows  f ( x2 ) > f ( x1 ), then a function is called increasing; if for any  x1 and x2  from the condition x2 > x1 it follows  f ( x2 ) < f ( x1 ), then a function is called decreasing.A function, which only increases or only decreases, is called a monotone function.

Bounded and unbounded functions. A function is bounded, if such positive number M exists, that |  f ( x ) | for all values of  x . If such positive number does not exist, then this function is unbounded.

 E x a m p l e s.

A function, shown on Fig.3, is a bounded, but not monotone function. On Fig.4 quite the opposite, we see a monotone, but unbounded function. ( Explain this, please ! ).

Continuous and discontinuous functions. A function y = f ( x ) is called a continuous function at a point x = a,  if:
 1)  the function is defined at  x = a , i.e.  f ( a ) exists;
 2)  a finite  lim  f ( x )  exists;
                    x a
( see the paragraph "Limits of functions" in the section “Principles of analysis”)

 3)   f ( a ) = lim  f ( x ) .
                    x a

If even one from these conditions isn’t executed, this function is called discontinuous at the point  x = a.

If a function is continuous at all points of its domain, it is called a continuous function.

Even and odd functions. If for any x from a function domain:  f ( – x ) =  f ( x ), then this function is called even;
if  f ( – x ) = – f ( x ), then this function is called odd . A graph of an even function is symmetrical relatively  y-axis ( Fig.5 ), a graph of an odd function is symmetrical relatively the origin of coordinates ( Fig.6 ).

Periodic function. A function  f ( x ) is periodic, if such non-zero number T existsthat for any x from a function domain:
 f ( x + T ) =  f ( x ). The least such number is called a period of a function. All trigonometric functions are periodic.

E x a m p l e   1 .   Prove that  sin x  has a number 2as a period.

S o l u t i o n .       We  know, that  sin ( x+ 2n ) = sin x,  where  n = 0, ± 1, ± 2, …
                             Hence, adding  2n  to an argument of a sine doesn’t change its value.
                            Maybe another number with the such property exists ?
                             Assume, that  P is the such number, i.e. the equality:

sin ( x + P ) = sin x,

                             is valid for any value of  x. Then this is valid for  x = / 2 , i.e.

sin ( / 2 + P ) = sin / 2 = 1.

                             But  sin ( / 2 + P ) = cos P  according to the reduction formula.Then from
                             the two last expressions it follows, that cos P = 1, but we know, that this
                             equality is right only if  P = 2n. Because the least non-zero number of
                            2 is 2, this is a period of  sin x. It is proved analogously, that 2 is also
                             a period for  cos x .
                             Prove, please, that functions tan x and cot x have as a period.

E x a m p l e   2.    What number is a period for the function sin 2x ?

S o l u t i o n .        Consider

sin 2x = sin ( 2x + 2n ) = sin [ 2 ( x + n ) ].

                              We see, that adding n to an argument  x, doesn’t change the function value.
                              The least non-zero number of is  , so this is a period of sin 2x .

Zeros of function. An argument value, at which a function is equal to zero, is called a zero ( root ) of the function. It can be that a function has some zeros. For instance, the function  y = x ( x + 1 ) ( x – 3)  has the three zeros:  x = 0,   x = – 1,   x = 3 . Geometrically, a zero of a function is  x-coordinate of a point of intersection of the function graph and  x-axis. On Fig.7 a graph of a function with zeros  x = a , x = b and  x =is represented.

Asymptote. If a graph of a function unboundedly approaches to some straight line at itstaking off an origin of coordinates, then this straight line is called an asymptote.


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