Difference between revisions of "Domain of a Function"

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Examples:
 
Examples:
* Let <math>S</math> be a set of ordered pairs such that <math> S = \{(1,2), (2,3), (4, 7), (13, 9), (-20, 0)\}</math>. The domain is the set of all x values of <math>S</math>, so the domain is <math>\{-20, 1, 2, 4, 13\}</math>.
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* Let <math> S </math> be a set of ordered pairs such that <math> S = \{(1,2), (2,3), (4, 7), (13, 9), (-20, 0)\}</math>. The domain is the set of all x values of <math> S </math>, so the domain is <math> \{-20, 1, 2, 4, 13\} </math>.
 
* The domain of <math> g(x) = 1/x </math> is all real numbers EXCEPT 0, since 1/0 is not defined.
 
* The domain of <math> g(x) = 1/x </math> is all real numbers EXCEPT 0, since 1/0 is not defined.
* The domain of <math> h(x) = \sqrt{x} </math> is the set <math> [0, \inf) </math>, since <math> \sqrt{x} </math> is only defined when <math> x </math> is nonnegative (that is, when <math> x </math> is greater than or equal to 0).
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* The domain of <math> h(x) = \sqrt{x} </math> is <math> [0, \inf) </math>, since <math> \sqrt{x} </math> is only defined when <math> x </math> is nonnegative (that is, when <math> x </math> is greater than or equal to 0).
  
==Resources and Examples==
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==Resources==
* [https://en.wikipedia.org/wiki/Domain_of_a_function Domain of a Function], Wikipedia
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* [https://www.youtube.com/watch?v=Q3NWljhiSJg Domain and Range: Basic Idea], patrickJMT
 
* [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-the-domain-of-a-function-defined-by-an-equation/ Finding Domain with an Equation], Lumen Learning
 
* [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-the-domain-of-a-function-defined-by-an-equation/ Finding Domain with an Equation], Lumen Learning
 
* [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-domain-and-range-from-graphs/ Finding Domain and Range with Graphs], Lumen Learning
 
* [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-domain-and-range-from-graphs/ Finding Domain and Range with Graphs], Lumen Learning
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* [https://www.youtube.com/watch?v=w81y25anEOM Finding the Domain Algebraically], patrickJMT
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* [https://www.youtube.com/watch?v=BxaYyS6lsQ4 Finding Domain and Range of a Piecewise Function], patrickJMT
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* [https://mathculus.com/how-to-find-the-domain-of-a-function-algebraically/ How to Find Domain + Example Problems], Math Culus
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== Licensing ==
 +
Content obtained and/or adapted from:
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* [https://en.wikipedia.org/wiki/Domain_of_a_function Domain of a Function, Wikipedia] under a CC BY-SA license

Latest revision as of 12:45, 21 October 2021

Definition

In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. It is the set X in the notation f: X → Y, and is alternatively denoted as dom(f). Since a (total) function is defined on its entire domain, its domain coincides with its domain of definition. However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain.

A domain is part of a function f if f is defined as a triple (X, Y, G), where X is called the domain of f, Y its codomain, and G its graph.

A domain is not part of a function f if f is defined as just a graph. For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (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 f: X → Y.

For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases).

If the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the x-axis. The domain of a function f can be thought of as the set of all x values that can be plugged into f(x) that return a valid output. For example, if we have a function g(x) in the Cartesian plane, the domain is all of the x values such that g(x) is a real number.

Examples:

  • Let be a set of ordered pairs such that . The domain is the set of all x values of , so the domain is .
  • The domain of is all real numbers EXCEPT 0, since 1/0 is not defined.
  • The domain of is , since is only defined when is nonnegative (that is, when is greater than or equal to 0).

Resources

Licensing

Content obtained and/or adapted from: