Difference between revisions of "Range"
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| − | In [[mathematics]], the '''range''' of a function are the [[value]]s it hits. | + | In [[mathematics]], the '''range''' (or '''image''') of a [[function]] are the [[value]]s it hits. It is not to be confused with the codomain of a function, which is a designated set to which all the values of the function belong. |
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| + | A function is [[onto]] (or surjective) if every value in its codomain is hit by the function, or, equivalently, if its range is equal to its codomain. More formally, a function <math>f: A \to B</math> is onto if for every <math>y \in B</math> there exists <math>x \in A</math> such that | ||
| + | <math>f(x) = y</math>. | ||
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| + | ==Examples== | ||
| + | Let <math>f: \mathbb{R} \to \mathbb{R}</math> be the function defined by the equation <math>f(x) = x^2</math>. By definition, the codomain | ||
| + | of <math>f</math> is <math>\mathbb{R}</math>. However, the range of <math>f</math> consists of all nonnegative real numbers. Indeed, let <math>y</math> be a nonnegative real number. Then <math>f(\sqrt{y}) = y</math>, and so <math>y</math> is one of the values hit by <math>f</math>. | ||
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| + | Let <math>g: \mathbb{R} \to \mathbb{R}</math> be the function defined by the equation <math>g(x) = x + 1</math>. Then, for every real number | ||
| + | <math>y</math>, we can see that <math>g(y-1) = (y-1) + 1 = y</math>, so every real number is hit by <math>g</math>. This means that the codomain and range of <math>g</math> are equal, namely <math>\mathbb{R}</math>. Therefore, <math>g</math> is onto. | ||
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| + | ==Non-mathematical uses== | ||
A range can also refer to a type of oven. | A range can also refer to a type of oven. | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 18:08, February 23, 2012
In mathematics, the range (or image) of a function are the values it hits. It is not to be confused with the codomain of a function, which is a designated set to which all the values of the function belong.
A function is onto (or surjective) if every value in its codomain is hit by the function, or, equivalently, if its range is equal to its codomain. More formally, a function is onto if for every there exists such that .
Examples
Let be the function defined by the equation . By definition, the codomain of is . However, the range of consists of all nonnegative real numbers. Indeed, let be a nonnegative real number. Then , and so is one of the values hit by .
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g: \mathbb{R} \to \mathbb{R}} be the function defined by the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = x + 1} . Then, for every real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , we can see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(y-1) = (y-1) + 1 = y} , so every real number is hit by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} . This means that the codomain and range of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} are equal, namely Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} . Therefore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is onto.
Non-mathematical uses
A range can also refer to a type of oven.