Difference between revisions of "Range of a Function"

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(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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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 range is the set of all y values of <math>S</math>, so the range is <math>\{0, 2, 3, 7, 9\}</math>.
 
* 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>.
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* 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.
 
* 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==
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==Resources==
 
* [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
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* [https://www.youtube.com/watch?v=Q3NWljhiSJg Domain and Range: Basic Idea], patrickJMT
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* [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=BxaYyS6lsQ4 Finding Domain and Range of a Piecewise Function], patrickJMT
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* [https://mathculus.com/how-to-find-the-range-of-a-function-algebraically/ How to Find Range + Example Problems], Math Culus
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== Licensing ==
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Content obtained and/or adapted from:
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* [https://en.wikipedia.org/wiki/Range_of_a_function Range of a function, Wikipedia] under a CC BY-SA license

Latest revision as of 12:47, 21 October 2021

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 be a set of ordered pairs such that . The range is the set of all y values of , so the range is .
  • The range of 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 (since the domain of this function is all nonzero numbers). So, we know that is in the range, where M is all nonzero numbers. There is no real number M such that though, which is why 0 is not in the range of .
  • The range of is . We can see this on the graph of 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

Licensing

Content obtained and/or adapted from: