FUNCTIONS OF ONE REAL VARIABLE

Transcription

1 FUNCTIONS OF ONE REAL VARIABLE

2 Concept o a unction Real unction o one real variable is a mapping rom the set M, a subset in real numbers R, to the set o all real numbers R. Function is a rule, by which any real number rom set M R can be attached eactly one real number y =. : M R, y Number M is independent variable - argument o a unction, number y M is dependent variable value o a unction. Set M = D is called domain o deinition o a unction, unction is deined on the set M. Set o all values o a unction is denoted H.

3 Formula y = Dierent deinitions o unction D is set o all such real number, or which the ormula has a real meaning. Table o values Values o unction in selected important values o variable are given eplicitly Word description o the relation Graph

4 Graph o a unction Set o all points [, y] in the plane with deined Cartesian orthogonal coordinate system Oy, or which D and y = is called graph o unction. G, y R : D y Curve passing through all points [, ]. A set o points in plane is graph o a unction, i each straight line parallel to y-ais has at most one common point with it. D is orthogonal projection o G onto -ais and H is orthogonal projection o G onto y-ais.

5 Operations on unctions 1. Absolute value o a unction is unction h deined on M and such, that or all M holds, : y R M, : 0 h R M h h

6 Operations on unctions. Product o a real number k and unction is unction h deined on M and such, that or all M holds, : y R M, : h R M h. k h

7 Equality o unctions 3. Functions and g are equal i and only i they are deined on the same set o real numbers D = Dg and or any rom this domain o deinition holds h

8 Sum, dierence and product o two unctions 4. Let two real unctions be given deined on the set M 1 R g deined on the set M R. Function h, with the domain o deinition M = M 1 M is called sum o unctions, i or all M holds h g dierence o unctions, i or all M holds h g product o unctions, i or all M holds h. g

9 Quotient o two unctions 5. Let two unctions be given deined on the set M 1 R g deined on the set M R. Let M M 1 M be the set o all such real numbers, or which the inequality holds: g 0. Function h, with the domain o deinition M is called quotient o unctions, i or all M holds h g

10 Composite unctions Function h =. g is called a composite unction composed rom unctions and g i and only i domain o deinition o unction h is set o all such numbers rom the domain o deinition Dg o unction g, in which the value o unction g is a number rom the domain o deinition D o unction and or any number M holds h = [g] Value o unction h in number equals to the value o unction u in number u = g. Function u is the major outside component, unction u = g is the minoir inside component o the composite unction h.

11 One-to-one unction Function deined on the set D is denoted as one-to-one unction, i or any two numbers 1, rom D holds: 1 1 A unction is one-to-one i each straight line parallel to -ais has at most one common point with G.

12 Inverse unction Let unction be deined on the set D and its set o values be H. Function -1 deined on the set H, with values in the set D is called inverse unction to the unction, i or any number b rom H holds: 1 b a a b Graphs o inverse unctions and -1 are symmetric with respect to the line = y.

13 Bounded unction Function deined on the set D is called bounded bounded above, bounded below, i there eists a real number K such, that or all rom D holds: K K, K It means, that a unction is bounded bounded above, bounded below, i its range H is a bounded bounded above, bounded below set o real numbers. Function that is not bounded is called unbounded unction.

14 Monotone unctions Function deined on the set M is called increasing, i decreasing, i non-decreasing, i non-increasing, i The above unctions are said to be monotone on M, incresing and decresing unctions are said to be strictly monotone.,, M,, M,, M,, M

15 Even and odd unctions Let unction be deined on the set M, which contains with any number also number. Function is said to be even on M, i M Function is said to be odd on M, i M Graph o an even unction is symmetric with respect to the y-ais, while graph o an odd unction is symmetric with respect to the origin O o the coordinate system.

16 Periodic unction Let unction be deined on the set M and p be a positive real number. Function is called periodic with the period p, i 1. or any p M also number p M. or all M holds p