# 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.

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

**. 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**

*R***of all admissible real values of an argument**

*X**x*, at which a function

*y*=

*f*(

*x*) is defined, is called a

*domain of*

*a function*. A set

**of all real values**

*Y**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*. From this definition it follows, that a function is given if :

**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– the domain of a function

**is given;**

*X*– the codomain of a function

**is given;**

*Y*– 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

*x*

_{1}and

*x*

_{2}from the condition

*x*

_{2 }>

*x*

_{1 }it follows

*f*(

*x*

_{2}

*) >*

*f*(

*x*

_{1 }), then a function is called

*increasing*; if for any

*x*

_{1}and

*x*

_{2}from the condition

*x*

_{2}

*>*

*x*

_{1}it follows

*f*(

*x*

_{2}

*) f (*

*x*

_{1}

*), 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*) |

*M*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+ *2*n *) = sin *x*, where *n* = 0, ± 1, ± 2, …

Hence, adding 2*n * 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:

*x +*

*P*) = sin

*x*,

is valid for any value of

*x*. Then this is valid for

*x*= / 2 , i.e.

*/ 2*

*+*

*P*) = sin / 2 = 1.

*/ 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*= 2

*n*. Because the least non-zero number of

2

*n*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 2*x* ?

S o l u t i o n . Consider

*x*= sin ( 2

*x +*2

*n*) = 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

*n*is , so this is a period of sin 2

*x*.

** 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*=

*c*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*.