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In mathematics, and more specifically in naive set theory, the range of a function refers to either the ''codomain'' or the ''image'' of the function, depending upon usage. Modern usage almost always uses ''range'' to mean ''image''. The codomain of a function is some arbitrary set. In real analysis, it is the real numbers. In complex analysis, it is the complex numbers. The image of a function is the set of all outputs of the function. The image is always a subset of the codomain. ==Distinguishing between the two uses== As the term "range" can have different meanings, it is considered a good practice to define it the first time it is used in a textbook or article. Older books, when they use the word "range", tend to use it to mean what is now called the codomain.〔Hungerford 1974, page 3.〕〔Childs 1990, page 140.〕 More modern books, if they use the word "range" at all, generally use it to mean what is now called the image.〔Dummit and Foote 2004, page 2.〕 To avoid any confusion, a number of modern books don't use the word "range" at all.〔Rudin 1991, page 99.〕 As an example of the two different usages, consider the function as it is used in real analysis, that is, as a function that inputs a real number and outputs its square. In this case, its codomain is the set of real numbers , but its image is the set of non-negative real numbers , since is never negative if is real. For this function, if we use "range" to mean ''codomain'', it refers to . When we use "range" to mean ''image'', it refers to . As an example where the range equals the codomain, consider the function , which inputs a real number and outputs its double. For this function, the codomain and the image are the same (the function is a surjection), so the word range is unambiguous; it is the set of all real numbers. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Range (mathematics)」の詳細全文を読む スポンサード リンク
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