{"id":6976,"date":"2021-07-23T23:19:25","date_gmt":"2021-07-23T14:19:25","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6976"},"modified":"2021-08-21T18:31:56","modified_gmt":"2021-08-21T09:31:56","slug":"functions","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/functions\/","title":{"rendered":"\ud568\uc218"},"content":{"rendered":"<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\\(S\\)\uc640 \\(T\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f:S \\rightarrow T\\)\uac00 <span class=\"defined\">\ud568\uc218<\/span>(function)\ub77c \ud568\uc740 \\(S\\)\uc758 \uac01 \uc6d0\uc18c \\(s\\)\ub97c \ube60\uc9d0 \uc5c6\uc774 \\(T\\)\uc758 \ud558\ub098\uc758 \uc6d0\uc18c \\(t\\)\uc5d0 \ub300\uc751\uc2dc\ud0a8 \uac83\uc774\ub2e4. \uc5ec\uae30\uc11c \\(s\\)\uc5d0 \ub300\uc751\ub41c \uc6d0\uc18c\ub97c \\(f\\)\uc5d0 \uc758\ud55c \\(s\\)\uc758 \ud568\uc22b\uac12\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(f(s)\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \ud568\uc218 \\(f:S \\rightarrow T\\)\ub97c \uac04\ub2e8\ud788 \\(f\\)\ub77c\uace0 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \uc9d1\ud569 \\(S\\)\ub97c \\(f\\)\uc758 <span class=\"defined\">\uc815\uc758\uc5ed<\/span>(domain)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \uc9d1\ud569 \\(T\\)\ub97c \\(f\\)\uc758 <span class=\"defined\">\uacf5\uc5ed<\/span>(codomain)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ud568\uc218 \\(f : S \\rightarrow T\\)\uc758 <span class=\"defined\">\uce58\uc5ed<\/span>(range, image)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<br \/>\n\\[\\operatorname{ran} (f) = \\operatorname{Im} (f) = \\left\\{ t\\in T \\,\\vert\\, \\exists s \\in S \\,:\\,\\, t=f(s) \\right\\}.\\]<br \/>\n\\(f : S \\rightarrow T\\)\uc640  \\(g : T \\rightarrow U\\)\uac00 \ud568\uc218\uc77c \ub54c, <span class=\"defined\">\ud569\uc131\ud568\uc218<\/span>(composition) \\(g \\circ f : S \\rightarrow U\\)\ub97c<br \/>\n\\[(g\\circ f)(s) = g(f(s)) \\quad\\text{for all}\\,\\,\\, s\\in S\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\uc778 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 0.2.1. (\ud568\uc218\uc758 \ud569\uc131\uc758 \uacb0\ud569 \uc131\uc9c8)<\/span><\/p>\n<p>\uc138 \ud568\uc218 \\(f : S \\rightarrow T ,\\) \\(g : T \\rightarrow U ,\\) \\(h : U \\rightarrow V\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[h\\circ (g \\circ f) = (h\\circ g)\\circ f.\\]<br \/>\n\uc989 \uc138 \ud568\uc218\uc758 \ud569\uc131\uc774 \uc815\uc758\ub420 \ub54c, \ud569\uc131\ud558\ub294 \uc21c\uc11c\uc5d0 \uc0c1\uad00 \uc5c6\uc774 \uadf8 \uacb0\uacfc\ub294 \ub3d9\uc77c\ud55c \ud568\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\\(f:S\\rightarrow T\\)\uac00 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n<ul>\n<li>\\(S\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c \\(x_1 ,\\) \\(x_2\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[x_1 \\ne x_2 \\quad\\Longrightarrow\\quad f\\left(x_1 \\right) \\ne f\\left(x_2 \\right)\\]<br \/>\n\uac00 \uc131\ub9bd\ud558\uba74, \\(f\\)\ub97c <span class=\"defined\">\uc77c\ub300\uc77c \ud568\uc218<\/span>(one-to-one) \ub610\ub294 <span class=\"defined\">\ub2e8\uc0ac\ud568\uc218<\/span>(injective function)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\\(\\operatorname{ran}(f) = T\\)\uc774\uba74, \\(f\\)\ub97c <span class=\"defined\">\uc704\ub85c\uc758 \ud568\uc218<\/span>(onto function) \ub610\ub294 <span class=\"defined\">\uc804\uc0ac\ud568\uc218<\/span>(surjective function)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<li>\\(f\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc774\uba74\uc11c \uc704\ub85c\uc758 \ud568\uc218\uc774\uba74, \\(f\\)\ub97c <span class=\"defined\">\uc77c\ub300\uc77c \ub300\uc751<\/span>(one-to-one correspondence) \ub610\ub294 <span class=\"defined\">\uc804\ub2e8\uc0ac\ud568\uc218<\/span>(bijective function)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/li>\n<\/ul>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 0.2.2.<\/span><\/p>\n<p>\ud568\uc218 \\(f:S\\rightarrow T\\)\uc640 \\(g:T\\rightarrow U\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \\(f\\)\uc640 \\(g\\)\uac00 \ubaa8\ub450 \uc77c\ub300\uc77c \ud568\uc218\uc774\uba74 \\(g \\circ f\\)\ub3c4 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(f\\)\uc640 \\(g\\)\uac00 \ubaa8\ub450 \uc704\ub85c\uc758 \ud568\uc218\uc774\uba74 \\(g \\circ f\\)\ub3c4 \uc704\ub85c\uc758 \ud568\uc218\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(f\\)\uc640 \\(g\\)\uac00 \ubaa8\ub450 \uc77c\ub300\uc77c \ub300\uc751\uc774\uba74 \\(g \\circ f\\)\ub3c4 \uc77c\ub300\uc77c \ub300\uc751\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(g \\circ f\\)\uac00 \uc77c\ub300\uc77c \ud568\uc218\uc774\uba74 \\(f\\)\ub3c4 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(g \\circ f\\)\uac00 \uc704\ub85c\uc758 \ud568\uc218\uc774\uba74 \\(g\\)\ub3c4 \uc704\ub85c\uc758 \ud568\uc218\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\\(f:S\\rightarrow T\\)\uac00 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \ud568\uc218 \\(g:T \\rightarrow S\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[g \\circ f = I_S \\quad\\text{and}\\quad f\\circ g = I_T\\]<br \/>\n\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\uba74, \\(f\\)\ub97c <span class=\"defined\">\uac00\uc5ed\ud568\uc218<\/span>(invertible function)\ub77c\uace0 \ubd80\ub974\uace0, \\(g\\)\ub97c \\(f\\)\uc758 <span class=\"defined\">\uc5ed\ud568\uc218<\/span>(inverse function)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \uc815\uc758\uc5d0\uc11c \\(I_S\\)\uc640 \\(I_T\\)\ub294<br \/>\n\\[\\begin{gather}<br \/>\nI_S (x) = x \\,\\,\\,\\text{for}\\,\\,\\, x\\in S ,\\\\[6pt]<br \/>\nI_T (y) = y \\,\\,\\,\\text{for}\\,\\,\\, y\\in T<br \/>\n\\end{gather}\\]<br \/>\n\ub85c \uc815\uc758\ub41c \ud56d\ub4f1\ud568\uc218\uc774\ub2e4.<\/p>\n<p>\uc5ed\ud568\uc218\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uba74, \\(g\\)\uac00 \\(f\\)\uc758 \uc5ed\ud568\uc218\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f\\)\uac00 \\(g\\)\uc758 \uc5ed\ud568\uc218\uc778 \uac83\uc774\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 0.2.3.<\/span><\/p>\n<p>\ud568\uc218 \\(f:S\\rightarrow T\\)\uac00 \uac00\uc5ed\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f:S\\rightarrow T\\)\uac00 \uc77c\ub300\uc77c \ub300\uc751\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/algebra-of-sets\">\uc9d1\ud569\uc758 \uc5f0\uc0b0<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/infinite-sets\">\uc720\ud55c\uc9d1\ud569\uacfc \ubb34\ud55c\uc9d1\ud569<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 0\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \\(S\\)\uc640 \\(T\\)\uac00 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f:S \\rightarrow T\\)\uac00 \ud568\uc218(function)\ub77c \ud568\uc740 \\(S\\)\uc758 \uac01 \uc6d0\uc18c \\(s\\)\ub97c \ube60\uc9d0 \uc5c6\uc774 \\(T\\)\uc758 \ud558\ub098\uc758 \uc6d0\uc18c \\(t\\)\uc5d0 \ub300\uc751\uc2dc\ud0a8 \uac83\uc774\ub2e4. \uc5ec\uae30\uc11c \\(s\\)\uc5d0 \ub300\uc751\ub41c \uc6d0\uc18c\ub97c \\(f\\)\uc5d0 \uc758\ud55c \\(s\\)\uc758 \ud568\uc22b\uac12\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(f(s)\\)\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \ud568\uc218 \\(f:S \\rightarrow T\\)\ub97c \uac04\ub2e8\ud788 \\(f\\)\ub77c\uace0 \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \uc9d1\ud569 \\(S\\)\ub97c \\(f\\)\uc758 \uc815\uc758\uc5ed(domain)\uc774\ub77c\uace0 \ubd80\ub974\uace0, \uc9d1\ud569 \\(T\\)\ub97c \\(f\\)\uc758 \uacf5\uc5ed(codomain)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud568\uc218&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":12,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6976","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6976","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=6976"}],"version-history":[{"count":19,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6976\/revisions"}],"predecessor-version":[{"id":7230,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6976\/revisions\/7230"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6976"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}