Difference between revisions of "Range"
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Let <math>A</math> and <math>B</math> be any [[set]]s and let <math>f:A\to B</math> be any [[function]] between them, so that <math>A</math> is the [[domain]] of <math>f</math> and <math>B</math> is the [[codomain]]. Then <math>\{b\in B\mid \mathrm{there\ is\ some\ } a\in A\mathrm{\ such\ that\ } f(a)=b\}</math> is called the '''range''' or '''image''' of <math>f</math>. | Let <math>A</math> and <math>B</math> be any [[set]]s and let <math>f:A\to B</math> be any [[function]] between them, so that <math>A</math> is the [[domain]] of <math>f</math> and <math>B</math> is the [[codomain]]. Then <math>\{b\in B\mid \mathrm{there\ is\ some\ } a\in A\mathrm{\ such\ that\ } f(a)=b\}</math> is called the '''range''' or '''image''' of <math>f</math>. | ||
| − | Thus, if we have <math>f: \mathbb{R} \to \mathbb{R}</math> given by <math>f(x) = x^2</math>, the range of <math>f</math> is the set of [[nonnegative]] | + | Thus, if we have <math>f: \mathbb{R} \to \mathbb{R}</math> given by <math>f(x) = x^2</math>, the range of <math>f</math> is the set of [[nonnegative]] [[real number]]s. |
A function is a [[surjection]] exactly when the range is equal to the codomain. | A function is a [[surjection]] exactly when the range is equal to the codomain. | ||
Revision as of 10:30, 31 July 2006
Let 
and 
be any sets and let 
be any function between them, so that is the domain of 
and is the codomain. Then 
is called the range or image of .
Thus, if we have 
given by 
, the range of is the set of nonnegative real numbers.
A function is a surjection exactly when the range is equal to the codomain.
