Function Definition Area

The domain of definition or the domain of the function is the set on which the function is defined. At each point of this set, the value of the function must be determined.

Content

Definition

If on the set $X$ ${\ displaystyle X}$ given a function that displays the set $X$ ${\ displaystyle X}$ into another set, then many $X$ ${\ displaystyle X}$ called the definition area or function definition area .

More formally, if a function is specified $f$ ${\ displaystyle f}$ which displays many $X$ ${\ displaystyle X}$ at $Y$ ${\ displaystyle Y}$ , i.e: $f\colon X\to Y$ ${\ displaystyle f \ colon X \ to Y}$ then the set $X$ ${\ displaystyle X}$ called the definition domain ^[1] or the task domain ^{[2] of the} function $f$ ${\ displaystyle f}$ and is designated $D(f)$ ${\ displaystyle D (f)}$ or $\mathrm {dom} \,f$ ${\ displaystyle \ mathrm {dom} \, f}$ (from English. domain - "area").

Sometimes functions defined on a subset are also considered. $D$ ${\ displaystyle D}$ some set $X$ ${\ displaystyle X}$ . In this case, many $X$ ${\ displaystyle X}$ called function sending area $f$ ${\ displaystyle f}$ ^[3] .

Examples

The most obvious examples of areas of definition are delivered by numerical functions . Measure and functionality also deliver the types of definition areas that are important in applications.

Numeric Functions

Numeric functions are functions that belong to the following two classes:

real-valued functions of a real variable are functions of the form $f\colon \mathbb {R} \to \mathbb {R}$ ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$ ;
as well as complex-valued functions of a complex variable form $f\colon \mathbb {C} \to \mathbb {C}$ ${\ displaystyle f \ colon \ mathbb {C} \ to \ mathbb {C}}$ ,

Where $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ and $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ - sets of real and complex numbers, respectively.

Identity Mapping

Function Definition Area $f(x)=x$ ${\ displaystyle f (x) = x}$ matches the dispatch area ( $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ or $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ )

Harmonic function

Function Definition Area $f(x)=1/x$ ${\ displaystyle f (x) = 1 / x}$ represents a complex plane without zero:

\mathrm {dom} \,f=\mathbb {C} \setminus \{0\}

{\ displaystyle \ mathrm {dom} \, f = \ mathbb {C} \ setminus \ {0 \}}

,

since the formula does not specify the value of the function at zero by any number, which is required in the formulation of the concept of a function. The departure area is the entire complex plane.

Fractional Rational Functions

The function definition domain of the form

f(x)={\frac {a_{0}+a_{1}x+\dots +a_{m}x^{m}}{b_{0}+b_{1}x+\dots +b_{n}x^{n}}}

{\ displaystyle f (x) = {\ frac {a_ {0} + a_ {1} x + \ dots + a_ {m} x ^ {m}} {b_ {0} + b_ {1} x + \ dots + b_ {n} x ^ {n}}}}

represents a real line or a complex plane with the exception of a finite number of points that are solutions of the equation

b_{0}+b_{1}x+\dots +b_{n}x^{n}=0

{\ displaystyle b_ {0} + b_ {1} x + \ dots + b_ {n} x ^ {n} = 0}

.

These points are called function poles . $f$ ${\ displaystyle f}$ .

For example, $f(x)={\frac {2x}{x^{2}-4}}$ ${\ displaystyle f (x) = {\ frac {2x} {x ^ {2} -4}}}$ defined at all points where the denominator does not vanish, that is, where $x^{2}-4\neq 0$ ${\ displaystyle x ^ {2} -4 \ neq 0}$ . In this way $\mathrm {dom} \,f$ ${\ displaystyle \ mathrm {dom} \, f}$ is the set of all real (or complex) numbers except 2 and -2.

Measure

If each point in the domain of definition of a function is a certain set, for example, a subset of a given set, then they say that a set function is given.

A measure is an example of such a function, where a certain set of subsets of a given set acts as the domain of definition of a function (measure), which is, for example, a ring or a semiring of sets.

For example, a certain integral is a function of the oriented span .

Functionality

Let be $\mathbb {F} =\{f\mid f\colon X\to \mathbb {R} \}$ ${\ displaystyle \ mathbb {F} = \ {f \ mid f \ colon X \ to \ mathbb {R} \}}$ - a family of mappings from the set $X$ ${\ displaystyle X}$ in many $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ . Then we can define a mapping of the form $F\colon \mathbb {F} \to \mathbb {R}$ ${\ displaystyle F \ colon \ mathbb {F} \ to \ mathbb {R}}$ . Such a mapping is called a functional .

If, for example, we fix some point $x_{0}\in ~X$ ${\ displaystyle x_ {0} \ in ~ X}$ , then you can define the function $F(f)=f(x_{0})$ ${\ displaystyle F (f) = f (x_ {0})}$ that takes in a “point” $f$ ${\ displaystyle f}$ same value as function itself $f$ ${\ displaystyle f}$ at the point $x_{0}$ ${\ displaystyle x_ {0}}$ .

Notes

↑ V. A. Sadovnichy . Theory of operators. - M .: Bustard, 2001 .-- S. 10 .-- 381 p. - ISBN 5-71-074297-X .
↑ V. A. Ilyin , V. A. Sadovnichy , Bl. H. Sendov . Chapter 3. Theory of limits // Mathematical analysis / Ed. A.N. Tikhonova . - 3rd ed. , reslave. and add. - M .: Prospect, 2006. - T. 1. - S. 105-121. - 672 p. - ISBN 5-482-00445-7 .
↑ V.A. Zorich . Chapter I. Some general mathematical concepts and notation. § 3. Function // Mathematical analysis. Part I. - The fourth, corrected. - M .: MCLMO, 2002 .-- S. 12-14. - 664 p. - ISBN 5-94057-056-9 .

Literature

Function, mathematical encyclopedic dictionary. - Ch. ed. Yu. V. Prokhorov. - M .: "Big Russian Encyclopedia", 1995.
Klein F. General concept of function . In the book: Elementary mathematics from the point of view of higher education. T.1. M.-L., 1933
I.A. Lavrov , L.L. Maksimova . Part I. Set theory // Problems in set theory, mathematical logic and theory of algorithms. - 3rd ed. . - M .: Fizmatlit, 1995 .-- S. 13 - 21. - 256 p. - ISBN 5-02-014844-X .
A.N. Kolmogorov , S.V. Fomin . Chapter 1 .. Elements of set theory // Elements of function theory and functional analysis. - 3rd ed. . - M .: Nauka, 1972. - S. 14 - 18. - 256 p.
J. L. Kelly . Chapter 0. Preliminaries // General Topology. - 2nd ed. . - M .: Nauka, 1981. - S. 19 - 27. - 423 p.
V.A. Zorich . Chapter I. Some general mathematical concepts and notation. § 3. Function // Mathematical analysis, part I. - M .: Nauka, 1981. - P. 23 - 36. - 544 p.
G.E. Shilov . Chapter 2. Elements of set theory. § 2.8. General concept of function. Graph // Mathematical analysis (functions of one variable). - M .: Nauka, 1969 .-- S. 65 - 69. - 528 p.
A.N. Kolmogorov . “What is a function” // “Quantum” . - M .: "Science" , 1970. - Vol. 1 . - S. 27-36 . - ISSN 0130-2221 .

[sadov-1] V. A. Sadovnichy . Theory of operators. - M .: Bustard, 2001 .-- S. 10 .-- 381 p. - ISBN 5-71-074297-X .

[ilyin-2] V. A. Ilyin , V. A. Sadovnichy , Bl. H. Sendov . Chapter 3. Theory of limits // Mathematical analysis / Ed. A.N. Tikhonova . - 3rd ed. , reslave. and add. - M .: Prospect, 2006. - T. 1. - S. 105-121. - 672 p. - ISBN 5-482-00445-7 .

[zorich-3] V.A. Zorich . Chapter I. Some general mathematical concepts and notation. § 3. Function // Mathematical analysis. Part I. - The fourth, corrected. - M .: MCLMO, 2002 .-- S. 12-14. - 664 p. - ISBN 5-94057-056-9 .