Basic notions and properties of functionsFunction. 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 ) | 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 ) .
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.sin ( x + P ) = sin x,
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:
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
2n 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 ?sin 2x = sin ( 2x + 2n ) = sin [ 2 ( x + n ) ].
S o l u t i o n . Consider
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 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 = 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.