Range of a Function - Revision history

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Resources and Examples

Lila at 23:38, 14 September 2021

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Resources and Examples

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Resources and Examples

Lila at 23:31, 14 September 2021

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Lila at 23:28, 14 September 2021

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Lila: Created page with "==Definition== In mathematics, the range of a function may refer to either of two closely related concepts: * The codomain of the function * The image of the function Given tw..."

2021-09-14T22:47:52Z

Created page with "==Definition== In mathematics, the range of a function may refer to either of two closely related concepts: * The codomain of the function * The image of the function Given tw..."

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==Definition==
In mathematics, the range of a function may refer to either of two closely related concepts:
* The codomain of the function
* The image of the function
Given two sets X and Y, a binary relation f between X and Y is a (total) function (from X to Y) if for every x in X there is exactly one y in Y such that f relates x to y. The sets X and Y are called domain and codomain of f, respectively. The image of f is then the subset of Y consisting of only those elements y of Y such that there is at least one x in X with f(x) = y.

In algebra, the range (or codomain) of a function is all of the possible outputs of the function. That is, if x is any element of the domain of some function f, then f(x) is in the range of the function f.

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 range is the set of all y values of <math>S</math>, so the range is <math>\{0, 2, 3, 7, 9\}</math>.
* The range of <math> g(x) = 1/x </math> is all real numbers EXCEPT for 0. We know this because for all nonzero real numbers M, 1/M is a nonzero number and is in the domain of <math> g(x) </math> (since, as seen in [[Domain of a Function]], the domain of this function is all nonzero numbers). So, we know that <math> 1/(1/M) = M </math> is in the range. There is no real number M such that <math> 1/M = 0 </math> though, which is why 0 is not in the range of <math> g(x) </math>.
* The range of <math> h(x) = x^2 + 2 </math> is <math> [2,\inf) </math>. We can see this on the graph of <math> h(x) </math> easily: the lowest point, or vertex, of the parabola is at (0, 2), so 2 is in the range. The parabola extends up to infinity on either side of the vertex, so we know that the range must be all numbers from 2 to infinity.
==Resources and Examples==
* [https://www.intmath.com/functions-and-graphs/2a-domain-and-range.php Domain and Range], Interactive Mathematics

← Older revision		Revision as of 18:47, 21 October 2021
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	* [https://www.youtube.com/watch?v=BxaYyS6lsQ4 Finding Domain and Range of a Piecewise Function], patrickJMT		* [https://www.youtube.com/watch?v=BxaYyS6lsQ4 Finding Domain and Range of a Piecewise Function], patrickJMT
	* [https://mathculus.com/how-to-find-the-range-of-a-function-algebraically/ How to Find Range + Example Problems], Math Culus		* [https://mathculus.com/how-to-find-the-range-of-a-function-algebraically/ How to Find Range + Example Problems], Math Culus
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Range_of_a_function Range of a function, Wikipedia] under a CC BY-SA license

@@ Line 15: / Line 15: @@
 * [https://www.intmath.com/functions-and-graphs/2a-domain-and-range.php Domain and Range], Interactive Mathematics
 * [https://www.youtube.com/watch?v=Q3NWljhiSJg Domain and Range: Basic Idea], patrickJMT
 * [https://www.youtube.com/watch?v=BxaYyS6lsQ4 Finding Domain and Range of a Piecewise Function], patrickJMT
 * [https://mathculus.com/how-to-find-the-range-of-a-function-algebraically/ How to Find Range + Example Problems], Math Culus

@@ Line 14: / Line 14: @@
 ==Resources and Examples==
 * [https://www.intmath.com/functions-and-graphs/2a-domain-and-range.php Domain and Range], Interactive Mathematics
 * [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-domain-and-range-from-graphs/ Finding Domain and Range with Graphs], Lumen Learning
 * [https://mathculus.com/how-to-find-the-range-of-a-function-algebraically/ How to Find Range + Example Problems], Math Culus

@@ Line 15: / Line 15: @@
 * [https://www.intmath.com/functions-and-graphs/2a-domain-and-range.php Domain and Range], Interactive Mathematics
 * [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-domain-and-range-from-graphs/ Finding Domain and Range with Graphs], Lumen Learning
 * [https://mathculus.com/how-to-find-the-range-of-a-function-algebraically/ How to Find Range + Example Problems], Math Culus

← Older revision		Revision as of 23:31, 14 September 2021
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	==Resources and Examples==		==Resources and Examples==
	* [https://www.intmath.com/functions-and-graphs/2a-domain-and-range.php Domain and Range], Interactive Mathematics		* [https://www.intmath.com/functions-and-graphs/2a-domain-and-range.php Domain and Range], Interactive Mathematics
		+	* [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-domain-and-range-from-graphs/ Finding Domain and Range with Graphs], Lumen Learning
		+	* [https://mathculus.com/how-to-find-the-range-of-a-function-algebraically/ How to Find Range + Example Problems], Math Culus

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 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 range is the set of all y values of <math>S</math>, so the range is <math>\{0, 2, 3, 7, 9\}</math>.
-* The range of <math> g(x) = 1/x </math> is all real numbers EXCEPT for 0. We know this because for all nonzero real numbers M, 1/M is a nonzero number and is in the domain of <math> g(x) </math> (since, as seen in [[Domain of a Function]], the domain of this function is all nonzero numbers). So, we know that <math> 1/(1/M) = M </math> is in the range. There is no real number M such that <math> 1/M = 0 </math> though, which is why 0 is not in the range of <math> g(x) </math>.
+* The range of <math> g(x) = 1/x </math> is all real numbers EXCEPT for 0. We know this because for all nonzero real numbers M, 1/M is a nonzero number and is in the domain of <math> g(x) </math> (since the domain of this function is all nonzero numbers). So, we know that <math> 1/(1/M) = M </math> is in the range, where M is all nonzero numbers. There is no real number M such that <math> 1/M = 0 </math> though, which is why 0 is not in the range of <math> g(x) </math>.
 * The range of <math> h(x) = x^2 + 2 </math> is <math> [2,\inf) </math>. We can see this on the graph of <math> h(x) </math> easily: the lowest point, or vertex, of the parabola is at (0, 2), so 2 is in the range. The parabola extends up to infinity on either side of the vertex, so we know that the range must be all numbers from 2 to infinity.
 ==Resources and Examples==
 * [https://www.intmath.com/functions-and-graphs/2a-domain-and-range.php Domain and Range], Interactive Mathematics