Difference between revisions of "Range of a Function"
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* 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 | + | ==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 | ||
+ | * [https://www.youtube.com/watch?v=Q3NWljhiSJg Domain and Range: Basic Idea], patrickJMT | ||
* [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 | ||
− | * [https://www.youtube.com/watch?v= | + | * [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 |
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
- Domain and Range, Interactive Mathematics
- Domain and Range: Basic Idea, patrickJMT
- Finding Domain and Range with Graphs, Lumen Learning
- Finding Domain and Range of a Piecewise Function, patrickJMT
- How to Find Range + Example Problems, Math Culus
Licensing
Content obtained and/or adapted from:
- Range of a function, Wikipedia under a CC BY-SA license