Domain - Revision history

Khanh at 23:58, 9 January 2022

2022-01-09T23:58:17Z

Khanh at 23:57, 9 January 2022

2022-01-09T23:57:47Z

Khanh at 23:56, 9 January 2022

2022-01-09T23:56:29Z

Khanh at 23:55, 9 January 2022

2022-01-09T23:55:56Z

Johnraymond.yanez: Added Links to materials

2020-08-17T14:12:27Z

Added Links to materials

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_Functions.pdf Domain Range and Toolkit Functions], Book Chapter
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_FunctionsGN.pdf Guided Notes]

@@ Line 1: / Line 1: @@
 [[File:Codomain2.SVG|right|thumb|250px|A function {{mvar|f}} from {{mvar|X}} to {{mvar|Y}}. The set of points in the red oval {{mvar|X}} is the domain of {{mvar|f}}.]]
-[[File:Square_root_0_25.svg|thumb|250px|Graph of the real-valued square root function, ''f''(''x'') = <math> \sqrt {x} </math>, whose domain consists of all nonnegative real numbers]]
+[[File:Square_root_0_25.svg|thumb|250px|Graph of the real-valued square root function, <math> f(x) = \sqrt {x} </math>, whose domain consists of all nonnegative real numbers]]
 In mathematics, the '''domain''' of a function is the set of inputs accepted by the function. It is sometimes denoted by <math>\operatorname{dom}(f)</math>, where {{math|''f''}} is the function.

@@ Line 1: / Line 1: @@
 [[File:Codomain2.SVG|right|thumb|250px|A function {{mvar|f}} from {{mvar|X}} to {{mvar|Y}}. The set of points in the red oval {{mvar|X}} is the domain of {{mvar|f}}.]]
-[[File:Square_root_0_25.svg|thumb|250px|Graph of the real-valued square root function, ''f''(''x'') = {{radic|''x''}}, whose domain consists of all nonnegative real numbers]]
+[[File:Square_root_0_25.svg|thumb|250px|Graph of the real-valued square root function, ''f''(''x'') = <math> \sqrt {x} </math>, whose domain consists of all nonnegative real numbers]]
 In mathematics, the '''domain''' of a function is the set of inputs accepted by the function. It is sometimes denoted by <math>\operatorname{dom}(f)</math>, where {{math|''f''}} is the function.
@@ Line 6: / Line 6: @@
 More precisely, given a function <math>f\colon X\to Y</math>, the domain of {{math|''f''}} is {{math|''X''}}. Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it.
-In the special case that {{math|''X''}} and {{math|''Y''}} are both subsets of <math>\R</math>, the function {{math|''f''}} can be graphed in the [[Cartesian coordinate system]]. In this case, the domain is represented on the {{math|''x''}}-axis of the graph, as the projection of the graph of the function onto the {{math|''x''}}-axis.
+In the special case that {{math|''X''}} and {{math|''Y''}} are both subsets of <math>\R</math>, the function {{math|''f''}} can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the {{math|''x''}}-axis of the graph, as the projection of the graph of the function onto the {{math|''x''}}-axis.
 For a function <math>f\colon X\to Y</math>, the set {{math|''Y''}} is called the codomain, and the set of values attained by the function (which is a subset of {{math|''Y''}}) is called its range or image.

← Older revision		Revision as of 23:56, 9 January 2022
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−	~~{{short description\|Mathematical concept}}~~
	[[File:Codomain2.SVG\|right\|thumb\|250px\|A function {{mvar\|f}} from {{mvar\|X}} to {{mvar\|Y}}. The set of points in the red oval {{mvar\|X}} is the domain of {{mvar\|f}}.]]		[[File:Codomain2.SVG\|right\|thumb\|250px\|A function {{mvar\|f}} from {{mvar\|X}} to {{mvar\|Y}}. The set of points in the red oval {{mvar\|X}} is the domain of {{mvar\|f}}.]]
	[[File:Square_root_0_25.svg\|thumb\|250px\|Graph of the real-valued square root function, ''f''(''x'') = {{radic\|''x''}}, whose domain consists of all nonnegative real numbers]]		[[File:Square_root_0_25.svg\|thumb\|250px\|Graph of the real-valued square root function, ''f''(''x'') = {{radic\|''x''}}, whose domain consists of all nonnegative real numbers]]

← Older revision		Revision as of 23:55, 9 January 2022
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		+	{{short description\|Mathematical concept}}
		+	[[File:Codomain2.SVG\|right\|thumb\|250px\|A function {{mvar\|f}} from {{mvar\|X}} to {{mvar\|Y}}. The set of points in the red oval {{mvar\|X}} is the domain of {{mvar\|f}}.]]
		+	[[File:Square_root_0_25.svg\|thumb\|250px\|Graph of the real-valued square root function, ''f''(''x'') = {{radic\|''x''}}, whose domain consists of all nonnegative real numbers]]
		+
		+	In mathematics, the '''domain''' of a function is the set of inputs accepted by the function. It is sometimes denoted by <math>\operatorname{dom}(f)</math>, where {{math\|''f''}} is the function.
		+
		+	More precisely, given a function <math>f\colon X\to Y</math>, the domain of {{math\|''f''}} is {{math\|''X''}}. Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it.
		+
		+	In the special case that {{math\|''X''}} and {{math\|''Y''}} are both subsets of <math>\R</math>, the function {{math\|''f''}} can be graphed in the [[Cartesian coordinate system]]. In this case, the domain is represented on the {{math\|''x''}}-axis of the graph, as the projection of the graph of the function onto the {{math\|''x''}}-axis.
		+
		+	For a function <math>f\colon X\to Y</math>, the set {{math\|''Y''}} is called the codomain, and the set of values attained by the function (which is a subset of {{math\|''Y''}}) is called its range or image.
		+
		+	Any function can be restricted to a subset of its domain. The restriction of <math>f \colon X \to Y</math> to <math>A</math>, where <math>A\subseteq X</math>, is written as <math>\left. f \right\|_A \colon A \to Y</math>.
		+
		+	== Natural domain ==
		+	If a real function {{mvar\|f}} is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function, and the set of real numbers on which the formula can be evaluated to a real number is called the '''natural domain''' or '''domain of definition''' of {{mvar\|f}}. In many contexts, a partial function is called simply a ''function'', and its natural domain is called simply its ''domain''.
		+
		+	=== Examples ===
		+
		+	* The function <math>f</math> defined by <math>f(x)=\frac{1}{x}</math> cannot be evaluated at 0. Therefore the natural domain of {{mvar\|f}} is <math>\mathbb{R} \setminus \{ 0 \}</math>.
		+	* In contrast, if <math>f</math> is the piecewise function <math>f(x) = \begin{cases}
		+	1/x&x\not=0\\
		+	0&x=0
		+	\end{cases},</math> then ''<math>f</math>'' is defined for all real numbers, and its natural domain is <math>\mathbb{R}</math>.
		+	* The function <math>x\mapsto\sqrt x</math> has as its natural domain the non-negative real numbers, which can be denoted by <math>\mathbb R_{\geq 0}</math>, by the interval <math>(0,\infty)</math>, or by <math>\{x\in\mathbb R:x\geq 0\}</math>.
		+	* The tangent function <math>\tan x</math> has as its natural domain the set of all real numbers which are not of the form <math>\tfrac{\pi}{2} + k \pi,</math> where {{mvar\|k}} is any integer.
		+
		+	== Other uses ==
		+	The word "domain" is used with other related meanings in some areas of mathematics. In topology, a domain is a connected open set. In real and complex analysis, a domain is an open connected subset of a real or complex vector space. In the study of partial differential equations, a domain is the open connected subset of the Euclidean space <math>\mathbb{R}^{n}</math> where a problem is posed (i.e., where the unknown function(s) are defined).
		+
		+	== Set theoretical notions ==
		+	For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class {{mvar\|X}}, in which case there is formally no such thing as a triple {{math\|(''X'', ''Y'', ''G'')}}. With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form {{math\|''f'': ''X'' → ''Y''}}.
		+
		+	== Resources ==
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_Functions.pdf Domain Range and Toolkit Functions], Book Chapter		* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_Functions.pdf Domain Range and Toolkit Functions], Book Chapter
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_FunctionsGN.pdf Guided Notes]		* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_FunctionsGN.pdf Guided Notes]
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Domain_of_a_function Domain of a function, Wikipedia] under a CC BY-SA license