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Difference between revisions of "Range of values"

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''of a function, set of values of a function''
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''of a function, set of values of a function, image of a function''
  
The set of all elements which a given function puts into correspondence with the elements of its domain of definition; that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077410/r0774101.png" />, then the set of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077410/r0774102.png" /> is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077410/r0774103.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077410/r0774104.png" /> for which there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077410/r0774105.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077410/r0774106.png" />. Thus, the range of values of a function is the image of its [[Domain of definition|domain of definition]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077410/r0774107.png" />.
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The set of all elements which a given function puts into correspondence with the elements of its domain of definition; that is, if $f : X \rightarrow Y$, then the set of values of $f$ is the set $Y_f$ of all $y \in Y$ for which there exists an $x \in X$ with $f(x) = y$. Thus, the range of values of a function is the image of its [[domain of definition]], $Y_f = f(X)$.
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See also [[Image of a morphism]].

Latest revision as of 21:17, 18 December 2014

of a function, set of values of a function, image of a function

The set of all elements which a given function puts into correspondence with the elements of its domain of definition; that is, if $f : X \rightarrow Y$, then the set of values of $f$ is the set $Y_f$ of all $y \in Y$ for which there exists an $x \in X$ with $f(x) = y$. Thus, the range of values of a function is the image of its domain of definition, $Y_f = f(X)$.

See also Image of a morphism.

How to Cite This Entry:
Range of values. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Range_of_values&oldid=12364
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article