[{"type":"paragraph","node_id":"lomkjatd1b2y","attrs":{},"children":[{"type":"text","text":"反函数"},{"type":"text","text":"（英文名：Inverse "},{"type":"internal_link","node_id":"lpf6zly45fo","attrs":{"docId":"7211828896328400952","target":1,"title":"函数","doc_id":"7211828896328400952"},"children":[{"type":"text","text":"Function"}]},{"type":"text","text":"），又称为逆函数"},{"type":"reference","node_id":"loy2rkb71as","attrs":{"content":"loy2rkb71as","data_index":0,"index":0}},{"type":"text","text":"，对一个定函数做逆运算的函数"},{"type":"reference","node_id":"lp6m1zo91q3y","attrs":{"content":"lp6m1zo91q3y","data_index":9,"index":9}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lpev8jvc21o4","attrs":{},"children":[{"type":"text","text":"反函数的定义是：设函数"},{"type":"formula","node_id":"lp6mgkky2jt","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239705494510632998","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2edc1192bea34b4293ade53c692bd2cb~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的定义域是"},{"type":"formula","node_id":"lp6mgkky1a8m","attrs":{"formula_height":"1.545","formula_width":"1.873","image_id":"7300942390863626291","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5edadcbaa0f14a15b708cd53e49e06c4~tplv-xv4ileqgde-image.image","value":"D"}},{"type":"text","text":"，"},{"type":"internal_link","node_id":"lqj1re4k1adr","attrs":{"docId":"6087720419503733950","target":1,"title":"值域","doc_id":"6087720419503733950"},"children":[{"type":"text","text":"值域"}]},{"type":"text","text":"是"},{"type":"formula","node_id":"lp6mgkkypol","attrs":{"formula_height":"1.545","formula_width":"2.378","image_id":"7226363506067505155","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a0dfff730d1147db971c497ed8ca7a0f~tplv-xv4ileqgde-image.image","value":"M"}},{"type":"text","text":"。如果对于值域任意的"},{"type":"formula","node_id":"lp6mgkky179i","attrs":{"formula_height":"2.009","formula_width":"6.252","image_id":"7302434522061275136","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f6fcc81b74a7452db3e0ffd2d6200d92~tplv-xv4ileqgde-image.image","value":"y\\in M"}},{"type":"text","text":"，通过关系式"},{"type":"formula","node_id":"lp6mgkky1zcz","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239705494510632998","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2edc1192bea34b4293ade53c692bd2cb~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"，都有唯一确定的数值"},{"type":"formula","node_id":"lp6mgkky4zr","attrs":{"formula_height":"1.636","formula_width":"5.933","image_id":"7294177925832753171","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/bd3ca4f598e64498b8180aa6c7607531~tplv-xv4ileqgde-image.image","value":"x\\in D"}},{"type":"text","text":"与之对应，那么由此所确定的以"},{"type":"formula","node_id":"lp6mgkky1e2d","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205608032565198900","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e546d1589ea34f2ab4e370532f6831d1~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"为自变量，"},{"type":"formula","node_id":"lp6mgkky1ip5","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7300957729076641827","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a1221d0b9a564f42aaa7c2ec2f43314f~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"为因变量的新函数叫做函数"},{"type":"formula","node_id":"lp6mgkkyu5y","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7294181781958459455","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0855dc981a464429afc02c92b2e9cbf5~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的反函数，记为"},{"type":"formula","node_id":"lp6mgkky19km","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7294266599006306343","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6bf1713a7e744ce48cf684fddf5d7a76~tplv-xv4ileqgde-image.image","value":"x=f^{-1}(y)"}},{"type":"text","text":"，它的定义域为"},{"type":"formula","node_id":"lp6mgkkys8f","attrs":{"formula_height":"1.545","formula_width":"2.378","image_id":"7226363506067505155","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a0dfff730d1147db971c497ed8ca7a0f~tplv-xv4ileqgde-image.image","value":"M"}},{"type":"text","text":"，"},{"type":"internal_link","node_id":"m0bjiw7o1q1k","attrs":{"docId":"6087720419503733950","target":1,"title":"值域","doc_id":"6087720419503733950"},"children":[{"type":"text","text":"值域"}]},{"type":"text","text":"为"},{"type":"formula","node_id":"lp6mgkky1psf","attrs":{"formula_height":"1.545","formula_width":"1.873","image_id":"7300942390863626291","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5edadcbaa0f14a15b708cd53e49e06c4~tplv-xv4ileqgde-image.image","value":"D"}},{"type":"reference","node_id":"lp6m4dpw91u","attrs":{"content":"lp6m4dpw91u","data_index":10,"index":10}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lp6m5v7u1okw","attrs":{},"children":[{"type":"text","text":"反函数"},{"type":"text","text":"的概念最早可以追溯到17世纪的"},{"type":"internal_link","node_id":"lp6mp16kogt","attrs":{"docId":"8530977079954154739","target":1,"title":"欧洲","doc_id":"8530977079954154739"},"children":[{"type":"text","text":"欧洲"}]},{"type":"text","text":"数学家阿德利昂·玛利·埃·勒让德（Adrien-Marie Legendre），他首先提出了函数的反函数的概念，并通过对称图形来描述反函数和原函数之间的关系"},{"type":"reference","node_id":"lp6m5x5gcil","attrs":{"content":"lp6m5x5gcil","data_index":1,"index":1}},{"type":"text","text":"。"},{"type":"text","text":"反函数"},{"type":"internal_link","node_id":"lpf6zly48y","attrs":{"docId":"1175255441073255932","target":1,"title":"存在性定理","doc_id":"1175255441073255932"},"children":[{"type":"text","text":"存在性定理"}]},{"type":"text","text":"为与其相关的重要定理，它的内容是：对于任意一个函数"},{"type":"formula","node_id":"loy31gok1ey0","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"来说，不一定有反函数。只有在函数的定义域与"},{"type":"internal_link","node_id":"lqj1re4r124x","attrs":{"docId":"6087720419503733950","target":1,"title":"值域","doc_id":"6087720419503733950"},"children":[{"type":"text","text":"值域"}]},{"type":"text","text":"之间建立了"},{"type":"internal_link","node_id":"lwu2eyhl4ra","attrs":{"docId":"7263049222198919204","target":1,"title":"双射","doc_id":"7263049222198919204"},"children":[{"type":"text","text":"一一映射"}]},{"type":"text","text":"关系的前提下，函数才存在"},{"type":"text","text":"反函数"},{"type":"reference","node_id":"loy2vbdn1gb8","attrs":{"content":"loy2vbdn1gb8","data_index":11,"index":11}},{"type":"text","text":"。反函数具有奇偶性、图像对称性、单调性等基本性质"},{"type":"reference","node_id":"loy2ubk7lzt","attrs":{"content":"loy2ubk7lzt","data_index":8,"index":8}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"loy2qo792w5","attrs":{},"children":[{"type":"text","text":"反函数的运算包括求一个函数的反函数"},{"type":"reference","node_id":"lp6mfbsu1wrd","attrs":{"content":"lp6mfbsu1wrd","data_index":12,"index":12}},{"type":"text","text":"、反函数求"},{"type":"internal_link","node_id":"lqj1re4k1qpc","attrs":{"docId":"385229163457989516","target":1,"title":"导数","doc_id":"385229163457989516"},"children":[{"type":"text","text":"导数"}]},{"type":"reference","node_id":"lp6mfope1oqr","attrs":{"content":"lp6mfope1oqr","data_index":13,"index":13}},{"type":"text","text":"、反函数求积分"},{"type":"reference","node_id":"lp6mg296a5f","attrs":{"content":"lp6mg296a5f","data_index":14,"index":14}},{"type":"text","text":"以及反函数的泰勒展开公式运算等"},{"type":"reference","node_id":"lp6mgbpfo4z","attrs":{"content":"lp6mgbpfo4z","data_index":15,"index":15}},{"type":"text","text":"。反函数在数学、物理、经济等领域都有应用，如运用反函数可以计算出一些函数的导数和积分"},{"type":"reference","node_id":"lp6m8bp919d3","attrs":{"content":"lp6m8bp919d3","data_index":3,"index":3}},{"type":"reference","node_id":"lp6m8bp9v2v","attrs":{"content":"lp6m8bp9v2v","data_index":4,"index":4}},{"type":"reference","node_id":"lp6m8bp91dnf","attrs":{"content":"lp6m8bp91dnf","data_index":5,"index":5}},{"type":"text","text":"。"}]},{"type":"heading","node_id":"1","attrs":{"level":1},"children":[{"type":"text","text":"定义"}]},{"type":"heading","node_id":"lp2qr01k1geu","attrs":{"level":2},"children":[{"type":"text","text":"映射"}]},{"type":"paragraph","node_id":"lp6dhnvtmum","attrs":{},"children":[{"type":"text","text":"设"},{"type":"formula","node_id":"lpf6ysyv1j44","attrs":{"formula_height":"2.149","formula_width":"5.916","image_id":"7305575550444240915","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ee8b65fac4344b98b423796bcc93b2ed~tplv-xv4ileqgde-image.image","value":"X、Y"}},{"type":"text","text":"是两个非空集合，如果存在一个法则"},{"type":"formula","node_id":"lpf6ysyv1kmw","attrs":{"formula_height":"2.059","formula_width":"1.244","image_id":"7205613778484854819","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/35513add2acb4a26b9bb80e2cb6fe06a~tplv-xv4ileqgde-image.image","value":"f"}},{"type":"text","text":"使得对"},{"type":"formula","node_id":"lpf6ysyvh8o","attrs":{"formula_height":"1.545","formula_width":"1.928","image_id":"7305575343035940927","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/36e553809563445fa20056d3dd76c85e~tplv-xv4ileqgde-image.image","value":"X"}},{"type":"text","text":"中每个元素"},{"type":"formula","node_id":"lpf6ysyv1nc","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7205530821682266123","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5d6a0958b3bb49afa5681fa8402e6f79~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"，按法则"},{"type":"formula","node_id":"lpf6ysyv12lo","attrs":{"formula_height":"2.059","formula_width":"1.244","image_id":"7205613778484854819","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/35513add2acb4a26b9bb80e2cb6fe06a~tplv-xv4ileqgde-image.image","value":"f"}},{"type":"text","text":"，在"},{"type":"formula","node_id":"lpf6ysyv1yc5","attrs":{"formula_height":"1.545","formula_width":"1.726","image_id":"7237513057226719284","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b759dcff44e34f6aac0b7f77242fe8de~tplv-xv4ileqgde-image.image","value":"Y"}},{"type":"text","text":"中有唯一确定的元素"},{"type":"formula","node_id":"lpf6ysyv1idu","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7305575239147192361","url":"//p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Y"}},{"type":"text","text":"。其中"},{"type":"formula","node_id":"lpf6ysyvcy1","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205526198313549862","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fdcdd22bc657443e9666202d8c51ebd6~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"称为元素"},{"type":"formula","node_id":"lpf6ysyv78l","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7305512946954895414","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/157d3fb660e84c89a3b291a922511b40~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"（在映射"},{"type":"formula","node_id":"lpf6ysyvgp","attrs":{"formula_height":"2.059","formula_width":"1.244","image_id":"7205613778484854819","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/35513add2acb4a26b9bb80e2cb6fe06a~tplv-xv4ileqgde-image.image","value":"f"}},{"type":"text","text":"下）的像，并记作"},{"type":"formula","node_id":"lpf6ysyvim5","attrs":{"formula_height":"2.262","formula_width":"4.299","image_id":"7305575012923342902","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5524be8120ef4d929b31405bdf69a6fe~tplv-xv4ileqgde-image.image","value":"f(x)"}},{"type":"text","text":"，即"},{"type":"formula","node_id":"lpf6ysyv207j","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239705494510632998","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2edc1192bea34b4293ade53c692bd2cb~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"，而元素"},{"type":"formula","node_id":"lpf6ysyv9fj","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7305575550444322835","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/05c7cfa1dcef47bc9c3d1ba78579e5f9~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"称为元素"},{"type":"formula","node_id":"lpf6ysyv171q","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205526198313549862","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fdcdd22bc657443e9666202d8c51ebd6~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"（在映射"},{"type":"formula","node_id":"lpf6ysyv3ix","attrs":{"formula_height":"2.059","formula_width":"1.244","image_id":"7305574228298940470","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d27f3c89f9be40469fa0a06147178996~tplv-xv4ileqgde-image.image","value":"f"}},{"type":"text","text":"下）的原像；集合"},{"type":"formula","node_id":"lpf6ysyv23no","attrs":{"formula_height":"1.545","formula_width":"1.928","image_id":"7305577299246678035","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7598dc45b4864eefbeccf2afa9fbbde3~tplv-xv4ileqgde-image.image","value":"X"}},{"type":"text","text":"称为映射"},{"type":"formula","node_id":"lpf6ysyv98e","attrs":{"formula_height":"2.059","formula_width":"1.244","image_id":"7201676703075139641","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2f1bf632bf3346cfa7401410104b4c0d~tplv-xv4ileqgde-image.image","value":"f"}},{"type":"text","text":"的定义域，记作"},{"type":"formula","node_id":"lpf6ysyv17qe","attrs":{"formula_height":"2.213","formula_width":"2.941","image_id":"7305574228298956854","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e49af0b6007948988669b6e954adca56~tplv-xv4ileqgde-image.image","value":"D_{f}"}},{"type":"text","text":"，即"},{"type":"formula","node_id":"lpf6ysyv1v69","attrs":{"formula_height":"1.545","formula_width":"1.928","image_id":"7305574228298891318","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/bb0031e528be4afd919a6560bd4a982d~tplv-xv4ileqgde-image.image","value":"X"}},{"type":"text","text":"中所有元素的像所组成的集合称为映射"},{"type":"formula","node_id":"lpf6ysyv1kyo","attrs":{"formula_height":"2.059","formula_width":"1.244","image_id":"7205613778484854819","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/35513add2acb4a26b9bb80e2cb6fe06a~tplv-xv4ileqgde-image.image","value":"f"}},{"type":"text","text":"的"},{"type":"internal_link","node_id":"lqj1re4m2gr","attrs":{"docId":"6087720419503733950","target":1,"title":"值域","doc_id":"6087720419503733950"},"children":[{"type":"text","text":"值域"}]},{"type":"text","text":"，记为"},{"type":"formula","node_id":"lpf6ysyv2d2","attrs":{"formula_height":"2.213","formula_width":"2.785","image_id":"7280806046765940797","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/588a101b411e4796a7207f38a0b2043b~tplv-xv4ileqgde-image.image","value":"R_{f}"}},{"type":"text","text":"或"},{"type":"formula","node_id":"lpf6ysyvo5q","attrs":{"formula_height":"2.262","formula_width":"4.932","image_id":"7305512946954928182","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3c25403e97d94801af46c36d45180b4f~tplv-xv4ileqgde-image.image","value":"f(X)"}},{"type":"text","text":"，即"},{"type":"formula","node_id":"lpf6ysyv1sve","attrs":{"formula_height":"2.364","formula_width":"26.929","image_id":"7305589923657023500","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/14d52695de4e474d892e102fa6276b64~tplv-xv4ileqgde-image.image","value":"R_{f}=f(X)=\\left\\{ f(x)| x\\in X\\right\\}"}},{"type":"reference","node_id":"lp6dpkzowzu","attrs":{"content":"lp6dpkzowzu","data_index":16,"index":16}},{"type":"reference","node_id":"lp0tkztb1tgx","attrs":{"content":"lp0tkztb1tgx","data_index":17,"index":17}},{"type":"text","text":"。"}]},{"type":"block_image","node_id":"m2347mgiz0i","attrs":{"url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d217e38fc7ec446bab23f105efac45f9~tplv-xv4ileqgde-image.image","alt":"","note":[{"type":"paragraph","node_id":"lpgivnmj1btz","attrs":{},"children":[{"type":"text","text":"映射"}]}],"align":"center","width":1210,"height":878,"id":"7424059744677249078","version":"1","license":"","name":"image.png","copyright":"no_copyright","layout_width":393,"layout_height":285,"quote_user_id":"","should_transform":false}},{"type":"heading","node_id":"lp2qtxgout9","attrs":{"level":2},"children":[{"type":"text","text":"函数"}]},{"type":"paragraph","node_id":"lomkjatdglp","attrs":{},"children":[{"type":"text","text":"设"},{"type":"formula","node_id":"lp6nhwkk1krc","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7300957729076641827","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a1221d0b9a564f42aaa7c2ec2f43314f~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"和"},{"type":"formula","node_id":"lp6nhwkk1n4n","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7241477165554860092","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8d4606b50ce1405ba840115e04012021~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"是两个变量，"},{"type":"formula","node_id":"lp6nhwkk1irt","attrs":{"formula_height":"1.545","formula_width":"1.873","image_id":"7300942390863626291","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5edadcbaa0f14a15b708cd53e49e06c4~tplv-xv4ileqgde-image.image","value":"D"}},{"type":"text","text":"为一个给定的非空数集，如果按照某个法则"},{"type":"formula","node_id":"lp6nhwkk313","attrs":{"formula_height":"2.059","formula_width":"1.244","image_id":"7221142792586657824","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/580b156e23ba4ff69fbc1db47a486090~tplv-xv4ileqgde-image.image","value":"f"}},{"type":"text","text":"，对每一个"},{"type":"formula","node_id":"lp6nhwkk24ar","attrs":{"formula_height":"1.636","formula_width":"5.933","image_id":"7240072304753901624","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d8398687a218418b90daa8f29ea30bdf~tplv-xv4ileqgde-image.image","value":"x\\in D"}},{"type":"text","text":"，变量"},{"type":"formula","node_id":"lp6nhwkke82","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7241477165554860092","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8d4606b50ce1405ba840115e04012021~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"总有唯一确定的数值与之对应，那么"},{"type":"formula","node_id":"lp6nhwkkfe8","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7241477165554860092","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8d4606b50ce1405ba840115e04012021~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"叫做"},{"type":"formula","node_id":"lp6nhwkkm9c","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7300957729076641827","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a1221d0b9a564f42aaa7c2ec2f43314f~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"的函数，记作"},{"type":"formula","node_id":"lp6nhwkkpro","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239705494510632998","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2edc1192bea34b4293ade53c692bd2cb~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"，"},{"type":"formula","node_id":"lp6nhwkkdda","attrs":{"formula_height":"1.636","formula_width":"5.933","image_id":"7240072304753901624","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d8398687a218418b90daa8f29ea30bdf~tplv-xv4ileqgde-image.image","value":"x\\in D"}},{"type":"text","text":"，其中变量"},{"type":"formula","node_id":"lp6nhwkk16h2","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7300957729076641827","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a1221d0b9a564f42aaa7c2ec2f43314f~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"叫做自变量，变量"},{"type":"formula","node_id":"lp6nhwkk13pw","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7241477165554860092","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8d4606b50ce1405ba840115e04012021~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"叫做函数或因变量，自变量的取值范围"},{"type":"formula","node_id":"lp6nhwkktxb","attrs":{"formula_height":"1.545","formula_width":"1.873","image_id":"7300942390863626291","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5edadcbaa0f14a15b708cd53e49e06c4~tplv-xv4ileqgde-image.image","value":"D"}},{"type":"text","text":"叫做函数的定义域，所有函数值组成的集合"},{"type":"formula","node_id":"lpf6ysyvnwt","attrs":{"formula_height":"2.262","formula_width":"26.008","image_id":"7305577299246809107","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e45c826678a3428f9599dc9f6779681c~tplv-xv4ileqgde-image.image","value":"W=\\left\\{ y|y=f(x) ，x\\in D\\right\\}"}},{"type":"text","text":"称为函数的"},{"type":"internal_link","node_id":"lqj1re4l1egb","attrs":{"docId":"6087720419503733950","target":1,"title":"值域","doc_id":"6087720419503733950"},"children":[{"type":"text","text":"值域"}]},{"type":"text","text":"。"},{"type":"formula","node_id":"lp6nhwkktna","attrs":{"formula_height":"2.059","formula_width":"1.244","image_id":"7221142792586657824","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/580b156e23ba4ff69fbc1db47a486090~tplv-xv4ileqgde-image.image","value":"f"}},{"type":"text","text":"是函数符号，它表示"},{"type":"formula","node_id":"lp6nhwkk1hqs","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7241477165554860092","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8d4606b50ce1405ba840115e04012021~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"与"},{"type":"formula","node_id":"lp6nhwkk1inl","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7300957729076641827","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a1221d0b9a564f42aaa7c2ec2f43314f~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"的对应规则，函数符号也可以用其他字母来表示"},{"type":"reference","node_id":"lp6nhsp31sip","attrs":{"content":"lp6nhsp31sip","data_index":18,"index":18}},{"type":"text","text":"。"}]},{"type":"heading","node_id":"lp2qpjb71oso","attrs":{"level":2},"children":[{"type":"text","text":"反函数及符号表示"}]},{"type":"paragraph","node_id":"lp6n2hy2kbf","attrs":{},"children":[{"type":"text","text":"设函数"},{"type":"formula","node_id":"lomq9j0k1431","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239705494510632998","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2edc1192bea34b4293ade53c692bd2cb~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的定义域是"},{"type":"formula","node_id":"lox1ly9f1dw0","attrs":{"formula_height":"1.545","formula_width":"1.873","image_id":"7300942390863626291","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5edadcbaa0f14a15b708cd53e49e06c4~tplv-xv4ileqgde-image.image","value":"D"}},{"type":"text","text":"，值域是"},{"type":"formula","node_id":"loxrtstj1vtn","attrs":{"formula_height":"1.545","formula_width":"2.378","image_id":"7226363506067505155","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a0dfff730d1147db971c497ed8ca7a0f~tplv-xv4ileqgde-image.image","value":"M"}},{"type":"text","text":"。如果对于"},{"type":"internal_link","node_id":"m0bjiw7l239t","attrs":{"docId":"6087720419503733950","target":1,"title":"值域","doc_id":"6087720419503733950"},"children":[{"type":"text","text":"值域"}]},{"type":"text","text":"任意的"},{"type":"formula","node_id":"lp2s5suo13jz","attrs":{"formula_height":"2.009","formula_width":"6.252","image_id":"7302434522061275136","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f6fcc81b74a7452db3e0ffd2d6200d92~tplv-xv4ileqgde-image.image","value":"y\\in M"}},{"type":"text","text":"，通过关系式"},{"type":"formula","node_id":"lp2s5suolip","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239705494510632998","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2edc1192bea34b4293ade53c692bd2cb~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"，都有唯一确定的数值"},{"type":"formula","node_id":"lp2s5suo10l9","attrs":{"formula_height":"1.636","formula_width":"5.933","image_id":"7294177925832753171","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/bd3ca4f598e64498b8180aa6c7607531~tplv-xv4ileqgde-image.image","value":"x\\in D"}},{"type":"text","text":"与之对应，那么由此所确定的以"},{"type":"formula","node_id":"lox1ly9fjgr","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205608032565198900","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e546d1589ea34f2ab4e370532f6831d1~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"为自变量，"},{"type":"formula","node_id":"lox1ly9f1ate","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7300957729076641827","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a1221d0b9a564f42aaa7c2ec2f43314f~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"为因变量的新函数叫做函数"},{"type":"formula","node_id":"lomq9j0k1xpz","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7294181781958459455","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0855dc981a464429afc02c92b2e9cbf5~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的反函数，其对应规律记作"},{"type":"formula","node_id":"lp2s5suou4e","attrs":{"formula_height":"2.464","formula_width":"3.597","image_id":"7302434539907907599","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/1e48acf80866473483c14d0b7c675bcd~tplv-xv4ileqgde-image.image","value":"f^{-1}"}},{"type":"text","text":"，"},{"type":"formula","node_id":"lp2s5suoc20","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239705494510632998","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2edc1192bea34b4293ade53c692bd2cb~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的反函数记为"},{"type":"formula","node_id":"lp2s5suo1u7u","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7294266599006306343","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6bf1713a7e744ce48cf684fddf5d7a76~tplv-xv4ileqgde-image.image","value":"x=f^{-1}(y)"}},{"type":"text","text":"，它的定义域为"},{"type":"formula","node_id":"lp2s5suo249f","attrs":{"formula_height":"1.545","formula_width":"2.378","image_id":"7226363506067505155","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a0dfff730d1147db971c497ed8ca7a0f~tplv-xv4ileqgde-image.image","value":"M"}},{"type":"text","text":"，值域为"},{"type":"formula","node_id":"lp2s5suo12rm","attrs":{"formula_height":"1.545","formula_width":"1.873","image_id":"7300942390863626291","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5edadcbaa0f14a15b708cd53e49e06c4~tplv-xv4ileqgde-image.image","value":"D"}},{"type":"text","text":"。原来的函数"},{"type":"formula","node_id":"lopl1ikkfyl","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239705494510632998","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2edc1192bea34b4293ade53c692bd2cb~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"称为原函数或直接函数"},{"type":"reference","node_id":"lp2s1uiu1dvo","attrs":{"content":"lp2s1uiu1dvo","data_index":10,"index":10}},{"type":"reference","node_id":"lomniygd13h3","attrs":{"content":"lomniygd13h3","data_index":2,"index":2}},{"type":"text","text":"。函数"},{"type":"formula","node_id":"lp0uas2zczq","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"中，字母"},{"type":"formula","node_id":"lp0uas2zmpd","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7234804544956268604","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/922051b23b6e4171876eea544bdf0ae5~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"表示自变量。若"},{"type":"formula","node_id":"lp0uas2zu51","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"存在"},{"type":"text","text":"反函数"},{"type":"text","text":"，为求反函数，需求以"},{"type":"formula","node_id":"lp0uas2z8nq","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7241477165554860092","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8d4606b50ce1405ba840115e04012021~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"为自变量表示"},{"type":"formula","node_id":"lp0uas2z1y32","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7234804544956268604","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/922051b23b6e4171876eea544bdf0ae5~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"："},{"type":"formula","node_id":"lp0uas2zdns","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7301936860853731337","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2db0a4b4afc04601bae80e8e0e618cb0~tplv-xv4ileqgde-image.image","value":"x=f^{-1}(y)"}},{"type":"reference","node_id":"lp0tvymw894","attrs":{"content":"lp0tvymw894","data_index":17,"index":17}},{"type":"text","text":"；习惯上，用字母"},{"type":"formula","node_id"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Wilhelm Leibniz）提出的。17世纪末，在他的文章中，首先使用了“"},{"type":"internal_link","node_id":"lpf6zly413jf","attrs":{"docId":"7211828896328400952","target":1,"title":"函数","doc_id":"7211828896328400952"},"children":[{"type":"text","text":"function"}]},{"type":"text","text":"”一词，翻译成汉语的意思就是“函数”。它指的是关于曲线上某点的一些"},{"type":"text","text":"线段"},{"type":"text","text":"的长，如“横坐标”“纵坐标”“弦”“"},{"type":"text","text":"切线"},{"type":"text","text":"”“"},{"type":"text","text":"法线"},{"type":"text","text":"”等概念。18世纪，"},{"type":"internal_link","node_id":"lp5qc883agv","attrs":{"docId":"","target":1,"title":"法国","doc_id":""},"children":[{"type":"text","text":"法国"}]},{"type":"text","text":"数学家"},{"type":"internal_link","node_id":"lpf6zly41iga","attrs":{"docId":"3677536811537581982","target":1,"title":"让·达朗贝尔","doc_id":"3677536811537581982"},"children":[{"type":"text","text":"让·勒朗·达朗贝尔"}]},{"type":"text","text":"在进行研究中，给函数重新下了一个定义，认为所谓变量的函数，就是指由这些变量和"},{"type":"text","text":"常量"},{"type":"text","text":"所组成的解析表达式，即用解析式表达函数关系。1837年，"},{"type":"internal_link","node_id":"lp5qc88o1dnd","attrs":{"docId":"7182166104352997432","target":1,"title":"德国","doc_id":"7182166104352997432"},"children":[{"type":"text","text":"德国"}]},{"type":"text","text":"数学家约翰·彼得·古斯塔夫·勒热纳·狄利克雷（Johann Peter Gustav Lejeune Dirichlet）进一步给出函数的定义为：对于在某区间上的每一个确定的"},{"type":"formula","node_id":"lp5oqmzd1xg6","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7205608108893143092","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7f836d985c704b729c48c5bdbdb1a8cf~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"值，"},{"type":"formula","node_id":"lp5oqmzd1whg","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205526198313549862","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fdcdd22bc657443e9666202d8c51ebd6~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"都有一个或多个确定的值，那么"},{"type":"formula","node_id":"lp5oqmzdma","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205526198313549862","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fdcdd22bc657443e9666202d8c51ebd6~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"叫做"},{"type":"formula","node_id":"lp5oqmzd1qpe","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7205608108893143092","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7f836d985c704b729c48c5bdbdb1a8cf~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"的函数"},{"type":"reference","node_id":"lp5gwoas1rp0","attrs":{"content":"lp5gwoas1rp0","data_index":22,"index":22}},{"type":"text","text":"。"}]},{"type":"block_image","node_id":"m23481gjdyl","attrs":{"url":"//p6-sdbk2-sign.byteimg.com/tos-cn-i-ry4let6mk6/b0a80576de9b488c93b7aa5d17280ad8~tplv-ry4let6mk6-image.image?lk3s=56f7592b&x-expires=1728640066&x-signature=C04aF94lMvzb1LIfzAxsWguLyP4%3D","alt":"","note":[{"type":"paragraph","node_id":"lp6mgkky3j","attrs":{},"children":[{"type":"text","text":"戈特弗里德·威廉·"},{"type":"internal_link","node_id":"lp5qc89e23za","attrs":{"docId":"7227487294625890319","target":1,"title":"戈特弗里德·莱布尼茨","doc_id":"7227487294625890319"},"children":[{"type":"text","text":"莱布尼茨"}]}]}],"align":"center","width":3818,"height":2342,"id":"7424056239698690059","version":"1","license":"","name":"2-gigapixel-width-1909px.png","copyright":"no_copyright","layout_width":280,"layout_height":171,"quote_user_id":"","should_transform":false}},{"type":"heading","node_id":"lp5lulu119fa","attrs":{"level":2},"children":[{"type":"text","text":"反函数概念的提出"}]},{"type":"paragraph","node_id":"lp6japbkyny","attrs":{},"children":[{"type":"text","text":"反函数的概念最早可以追溯到17世纪的"},{"type":"internal_link","node_id":"lp6mp16kukr","attrs":{"docId":"8530977079954154739","target":1,"title":"欧洲","doc_id":"8530977079954154739"},"children":[{"type":"text","text":"欧洲"}]},{"type":"text","text":"数学家阿德利昂·玛利·埃·勒让德，他首先提出了函数的反函数的概念，并通过对称图形来描述两个函数之间的关系"},{"type":"reference","node_id":"lp6l2td57uv","attrs":{"content":"lp6l2td57uv","data_index":1,"index":1}},{"type":"text","text":"。同一时期，"},{"type":"internal_link","node_id":"lp6mp16k1deo","attrs":{"docId":"7298181419314085927","target":1,"title":"英国","doc_id":"7298181419314085927"},"children":[{"type":"text","text":"英国"}]},{"type":"text","text":"物理学家、数学家"},{"type":"internal_link","node_id":"lp6mp16kaz3","attrs":{"docId":"7194722585664536632","target":1,"title":"艾萨克·牛顿","doc_id":"7194722585664536632"},"children":[{"type":"text","text":"艾萨克·牛顿"}]},{"type":"text","text":"（Isaac 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M"}},{"type":"text","text":"，则它的相反的映射"},{"type":"formula","node_id":"lp0uas30jlq","attrs":{"formula_height":"2.464","formula_width":"13.629","image_id":"7301937711408021541","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/deacf38283fe498cab70723053800747~tplv-xv4ileqgde-image.image","value":"f^{-1}：M\\rightarrow D"}},{"type":"text","text":"也为一一映射，成为一个函数（即得"},{"type":"formula","node_id":"lpf6ysyv34t","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"反函数"},{"type":"text","text":"）"},{"type":"reference","node_id":"lp0tkstn1229","attrs":{"content":"lp0tkstn1229","data_index":17,"index":17}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lp0tgtbh19pq","attrs":{},"children":[{"type":"text","text":"反函数的存在性表明：对于任意一个函数"},{"type":"formula","node_id":"loohrb6cgtw","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"来说，不一定都有反函数"},{"type":"reference","node_id":"lp0tcqc91x3l","attrs":{"content":"lp0tcqc91x3l","data_index":11,"index":11}},{"type":"text","text":"。如果函数"},{"type":"formula","node_id":"loohrb6c1kjm","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"有反函数"},{"type":"formula","node_id":"loohrb6c7z3","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7294268518063882294","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6c80c0d14edf430d9a2b684942c95496~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)"}},{"type":"text","text":"，那么原来函数"},{"type":"formula","node_id":"loohrb6cbzb","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"也是其反函数"},{"type":"formula","node_id":"loohrb6c1qw4","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7294268518063882294","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6c80c0d14edf430d9a2b684942c95496~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)"}},{"type":"text","text":"的反函数，即它们互为反函数"},{"type":"reference","node_id":"loofet6grp6","attrs":{"content":"loofet6grp6","data_index":11,"index":11}},{"type":"text","text":"。函数存在反函数的充要条件是对不同的自变量值，其函数值均不同。因此，要判定一个函数存在反函数，需判定对定义域内任意的两个自变量值，是否总有它们的函数值也不同，只需找出两个不同的自变量值，它们的函数值却相同"},{"type":"reference","node_id":"loxq0cwk6jh","attrs":{"content":"loxq0cwk6jh","data_index":12,"index":12}},{"type":"text","text":"。"}]},{"type":"heading","node_id":"3","attrs":{"level":1},"children":[{"type":"text","text":"性质"}]},{"type":"bullet_list","node_id":"lp2uhq421j33","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lompzjmo18vb","attrs":{},"children":[{"type":"text","text":"图像对称性"}]}]}]},{"type":"paragraph","node_id":"lp2uho1l1pvp","attrs":{},"children":[{"type":"text","text":"函数"},{"type":"formula","node_id":"lomq9j0kq4f","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7294181781958459455","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0855dc981a464429afc02c92b2e9cbf5~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"与它的反函数"},{"type":"formula","node_id":"lomq9j0k1l7l","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7298283665291935778","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/777cf2f29d324cfea42eb3bce4b3235e~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)"}},{"type":"text","text":"的图像关于直线"},{"type":"formula","node_id":"lomkj8nb1dvg","attrs":{"formula_height":"2.009","formula_width":"5.584","image_id":"7346098183027720246","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0603695b3eb44c4caad697ca3500c264~tplv-xv4ileqgde-image.image","value":"y=x"}},{"type":"text","text":"对称。若点"},{"type":"formula","node_id":"lqb43zkkkbc","attrs":{"formula_height":"2.262","formula_width":"6.19","image_id":"7241486455967580216","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f878c3afaa684ec7bdb713cc5bcb7c8e~tplv-xv4ileqgde-image.image","value":"(a，b)"}},{"type":"text","text":"在"},{"type":"formula","node_id":"lqb43zkkatb","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7294181781958459455","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0855dc981a464429afc02c92b2e9cbf5~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"图像上，则"},{"type":"formula","node_id":"lqb43zkk20f9","attrs":{"formula_height":"2.262","formula_width":"6.19","image_id":"7313938222172766223","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/1ad15bffcb3a439d89bd3dcd393f62d7~tplv-xv4ileqgde-image.image","value":"(b，a)"}},{"type":"text","text":"在"},{"type":"formula","node_id":"lqb43zkk100t","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7298283665291935778","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/777cf2f29d324cfea42eb3bce4b3235e~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)"}},{"type":"text","text":"图像上"},{"type":"reference","node_id":"lonsdkze108a","attrs":{"content":"lonsdkze108a","data_index":25,"index":25}},{"type":"reference","node_id":"lqb3rpwzia4","attrs":{"content":"lqb3rpwzia4","data_index":26,"index":26}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lqb3v1gy2512","attrs":{},"children":[{"type":"text","text":"例：若点"},{"type":"formula","node_id":"lqb43zkk7ws","attrs":{"formula_height":"2.262","formula_width":"6.285","image_id":"7313941400566022196","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/60997413eb764a6c8646d6407fe154ad~tplv-xv4ileqgde-image.image","value":"(1，2)"}},{"type":"text","text":"在函数"},{"type":"formula","node_id":"lqb43zkk185m","attrs":{"formula_height":"2.91","formula_width":"12.661","image_id":"7313940612665868288","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b4c9cd2dd30348e2a91b043a81da721d~tplv-xv4ileqgde-image.image","value":"y=\\sqrt{ax+b}"}},{"type":"text","text":"的图像上，由在它的反函数的图像上，求"},{"type":"formula","node_id":"lqb43zkk1djb","attrs":{"formula_height":"2.149","formula_width":"4.43","image_id":"7313938222172782607","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/25059b5461ca440ebd938bbb67faeefa~tplv-xv4ileqgde-image.image","value":"a，b"}},{"type":"text","text":"的值"},{"type":"reference","node_id":"lqb438qy5qd","attrs":{"content":"lqb438qy5qd","data_index":26,"index":26}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lqb3wo271dih","attrs":{},"children":[{"type":"text","text":"解：由点"},{"type":"formula","node_id":"lqb43zkkfdq","attrs":{"formula_height":"2.262","formula_width":"6.285","image_id":"7313941400566022196","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/60997413eb764a6c8646d6407fe154ad~tplv-xv4ileqgde-image.image","value":"(1，2)"}},{"type":"text","text":"在函数"},{"type":"formula","node_id":"lqb43zkk12pg","attrs":{"formula_height":"2.91","formula_width":"12.661","image_id":"7313940612665868288","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b4c9cd2dd30348e2a91b043a81da721d~tplv-xv4ileqgde-image.image","value":"y=\\sqrt{ax+b}"}},{"type":"text","text":"的图像上得"},{"type":"formula","node_id":"lqb43zkk5r8","attrs":{"formula_height":"2.851","formula_width":"11.389","image_id":"7313940786666684416","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/191cc72579bb4bb0b2a862e0d31471a2~tplv-xv4ileqgde-image.image","value":"2=\\sqrt{a+b}"}},{"type":"text","text":"，又在它的反函数图像上，则"},{"type":"formula","node_id":"lqb43zkk11o0","attrs":{"formula_height":"2.262","formula_width":"6.285","image_id":"7313940786666700800","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/cb93fc394ae1489283731089bcb086df~tplv-xv4ileqgde-image.image","value":"(2，1)"}},{"type":"text","text":"在函数"},{"type":"formula","node_id":"lqb43zkk81q","attrs":{"formula_height":"2.91","formula_width":"12.661","image_id":"7313940612665868288","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b4c9cd2dd30348e2a91b043a81da721d~tplv-xv4ileqgde-image.image","value":"y=\\sqrt{ax+b}"}},{"type":"text","text":"的图像上，得"},{"type":"formula","node_id":"lqb43zkk1sad","attrs":{"formula_height":"2.851","formula_width":"12.52","image_id":"7313943130020020276","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/18ef21ff952a416c8073883c3bbda6c5~tplv-xv4ileqgde-image.image","value":"1=\\sqrt{2a+b}"}},{"type":"text","text":"，二式联立解得"},{"type":"formula","node_id":"lqb43zkk1brc","attrs":{"formula_height":"2.149","formula_width":"14.487","image_id":"7313945636460576802","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f504399188be47f58f655b075c84f8ae~tplv-xv4ileqgde-image.image","value":"a=-3，b=7"}},{"type":"reference","node_id":"lqb43avv1zqp","attrs":{"content":"lqb43avv1zqp","data_index":26,"index":26}},{"type":"text","text":"。"}]},{"type":"block_image","node_id":"m2348pb021e5","attrs":{"url":"//p9-sdbk2-sign.byteimg.com/tos-cn-i-ry4let6mk6/1d279bba1be14f63a47cc9bcd667c3b9~tplv-ry4let6mk6-image.image?lk3s=56f7592b&x-expires=1728640097&x-signature=xGCs0i9HIG1F1qfAw%2Fu4OU2xmWk%3D","alt":"","note":[{"type":"paragraph","node_id":"lozl66iqa6i","attrs":{},"children":[{"type":"text","text":"反函数图像的对称性"}]}],"align":"center","width":892,"height":878,"id":"7424065714555928630","version":"1","license":"","name":"image.png","copyright":"no_copyright","layout_width":217,"layout_height":213,"quote_user_id":"","should_transform":false}},{"type":"bullet_list","node_id":"lp2uhm5hy5j","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lpewh4u9dcc","attrs":{},"children":[{"type":"text","text":"图像交点的性质"}]}]}]},{"type":"paragraph","node_id":"lpewh4u91cba","attrs":{},"children":[{"type":"text","text":"同一"},{"type":"internal_link","node_id":"lpf6zly44e1","attrs":{"docId":"5448778957468456611","target":1,"title":"坐标系","doc_id":"5448778957468456611"},"children":[{"type":"text","text":"坐标系"}]},{"type":"text","text":"内，函数"},{"type":"formula","node_id":"lpf6ysyv211a","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7294181781958459455","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0855dc981a464429afc02c92b2e9cbf5~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"与它的"},{"type":"text","text":"反函数"},{"type":"formula","node_id":"lpf6ysyvfhn","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7298283665291935778","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/777cf2f29d324cfea42eb3bce4b3235e~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)"}},{"type":"text","text":"的图像的交点，或者在直线"},{"type":"formula","node_id":"lpf6ysyv1jqa","attrs":{"formula_height":"1.783","formula_width":"5.42","image_id":"7242651112442953768","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f2107a61f0ed47de9da8e13ecafe2935~tplv-xv4ileqgde-image.image","value":"y=x"}},{"type":"text","text":"上，或者关于直线"},{"type":"formula","node_id":"lpf6ysyv1tkr","attrs":{"formula_height":"1.783","formula_width":"5.42","image_id":"7242651112442953768","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f2107a61f0ed47de9da8e13ecafe2935~tplv-xv4ileqgde-image.image","value":"y=x"}},{"type":"text","text":"对称地成对出现"},{"type":"reference","node_id":"lpewh4u6114z","attrs":{"content":"lpewh4u6114z","data_index":26,"index":26}},{"type":"reference","node_id":"lpewh4u624ps","attrs":{"content":"lpewh4u624ps","data_index":27,"index":27}},{"type":"text","text":"。若函数"},{"type":"formula","node_id":"lpf6ysyv228l","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"是单调增函数，那么"},{"type":"formula","node_id":"lpf6ysyv1hpc","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7298284850862260258","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/13cccf304e1d49019e5c379e0a41ab9b~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"与它的反函数"},{"type":"formula","node_id":"lpf6ysyv3h6","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7239711159299866681","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/864a35a6284d49ac86ba0fb224d0f5c5~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)"}},{"type":"text","text":"的图像的交点必定在直线"},{"type":"formula","node_id":"lpf6ysyv16jt","attrs":{"formula_height":"1.783","formula_width":"5.42","image_id":"7242651112442953768","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f2107a61f0ed47de9da8e13ecafe2935~tplv-xv4ileqgde-image.image","value":"y=x"}},{"type":"text","text":"上"},{"type":"reference","node_id":"lpewh4u68da","attrs":{"content":"lpewh4u68da","data_index":28,"index":28}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lqazl23fkqz","attrs":{},"children":[{"type":"text","text":"证明：若点"},{"type":"formula","node_id":"lqb307sw9uh","attrs":{"formula_height":"2.262","formula_width":"13.135","image_id":"7313908108487720995","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2d0438b42bac4ca8b41226165e5ce008~tplv-xv4ileqgde-image.image","value":"(a，b)(a\\ne b)"}},{"type":"text","text":"是"},{"type":"formula","node_id":"lqb307sw1hud","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7294181781958459455","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0855dc981a464429afc02c92b2e9cbf5~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"与"},{"type":"formula","node_id":"lqb307swjvz","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7298283665291935778","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/777cf2f29d324cfea42eb3bce4b3235e~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)"}},{"type":"text","text":"的图像的交点，则点"},{"type":"formula","node_id":"lqb307sw17if","attrs":{"formula_height":"2.262","formula_width":"6.19","image_id":"7313909190437879843","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/84fbfa0794814dd3bc5601bfe5aba70f~tplv-xv4ileqgde-image.image","value":"(b，a)"}},{"type":"text","text":"必定也是。所以"},{"type":"formula","node_id":"lqb307sw1ndw","attrs":{"formula_height":"2.262","formula_width":"8.189","image_id":"7313910086018105384","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/51df7cc2b69a469aac5e91e8d21cd530~tplv-xv4ileqgde-image.image","value":"b=f(a)"}},{"type":"text","text":"及"},{"type":"formula","node_id":"lqb307swets","attrs":{"formula_height":"2.262","formula_width":"8.189","image_id":"7313903054238302242","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/4831a3803e1142acab1e758433c495f0~tplv-xv4ileqgde-image.image","value":"a=f(b)"}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lqazmug61smo","attrs":{},"children":[{"type":"text","text":"因为"},{"type":"formula","node_id":"lqb307sw20yn","attrs":{"formula_height":"2.106","formula_width":"5.185","image_id":"7313904544956825600","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b82d58c53a584537ba1fb7049d0360b5~tplv-xv4ileqgde-image.image","value":"a\\ne b"}},{"type":"text","text":"，不妨设"},{"type":"formula","node_id":"lqb307sw1ne4","attrs":{"formula_height":"1.661","formula_width":"5.185","image_id":"7313912009517580288","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6dea677faf244a90ad7db87c6acf03e5~tplv-xv4ileqgde-image.image","value":"a>b"}},{"type":"text","text":"，又因为"},{"type":"formula","node_id":"lqb307swajz","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7294181781958459455","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0855dc981a464429afc02c92b2e9cbf5~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"是单调增函数，所以"},{"type":"formula","node_id":"lqb307swvbh","attrs":{"formula_height":"2.262","formula_width":"11.194","image_id":"7313915087243264034","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a29636366a384222a0c22a903eb81096~tplv-xv4ileqgde-image.image","value":"f(a)>f(b)"}},{"type":"text","text":"，因为"},{"type":"formula","node_id":"lqb307sw1gzb","attrs":{"formula_height":"2.262","formula_width":"18.641","image_id":"7313904547905404980","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a819ea9875a246b3b16b1b7dc99c02ac~tplv-xv4ileqgde-image.image","value":"b=f(a)、a=f(b)"}},{"type":"text","text":"，所以"},{"type":"formula","node_id":"lqb307sw1c5u","attrs":{"formula_height":"1.661","formula_width":"5.185","image_id":"7310610914406039587","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/55e4c51f854f43f4ac995a0e43fd7a32~tplv-xv4ileqgde-image.image","value":"b>a"}},{"type":"text","text":"，这就与"},{"type":"formula","node_id":"lqb307sw1myg","attrs":{"formula_height":"1.661","formula_width":"5.185","image_id":"7313912009517580288","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6dea677faf244a90ad7db87c6acf03e5~tplv-xv4ileqgde-image.image","value":"a>b"}},{"type":"text","text":"相矛盾了，因此只能"},{"type":"formula","node_id":"lqb307swmh4","attrs":{"formula_height":"1.756","formula_width":"5.185","image_id":"7313907984873226280","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5120d564dbb749ab9d0226c0c2e94ca6~tplv-xv4ileqgde-image.image","value":"a=b"}},{"type":"text","text":"，故交点必定在直线"},{"type":"formula","node_id":"lqb307sw12zj","attrs":{"formula_height":"1.783","formula_width":"5.42","image_id":"7242651112442953768","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f2107a61f0ed47de9da8e13ecafe2935~tplv-xv4ileqgde-image.image","value":"y=x"}},{"type":"text","text":"上"},{"type":"reference","node_id":"lqb07unhvc7","attrs":{"content":"lqb07unhvc7","data_index":28,"index":28}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lqb08z9a1we3","attrs":{},"children":[{"type":"text","text":"例：解"},{"type":"internal_link","node_id":"m0bjiw7j12jc","attrs":{"docId":"2293567924838015583","target":1,"title":"方程","doc_id":"2293567924838015583"},"children":[{"type":"text","text":"方程"}]},{"type":"formula","node_id":"lqb307swxf9","attrs":{"formula_height":"5.016","formula_width":"20.706","image_id":"7313908108487819299","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/4a2489dc6d344bae832bea75b0a1ffa1~tplv-xv4ileqgde-image.image","value":"\\sqrt[3]{5x-12}=\\frac{x^{3}+12}{5}"}},{"type":"reference","node_id":"lqb0sd5e1k25","attrs":{"content":"lqb0sd5e1k25","data_index":28,"index":28}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lqb0g97521sh","attrs":{},"children":[{"type":"text","text":"解：因为函数"},{"type":"formula","node_id":"lqb307sw1v9p","attrs":{"formula_height":"2.652","formula_width":"13.509","image_id":"7313927707903950863","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/663be80dd6a54ba2b6816b1038394225~tplv-xv4ileqgde-image.image","value":"y=\\sqrt[3]{5x-12}"}},{"type":"text","text":"是定义在"},{"type":"formula","node_id":"lqb307swd7p","attrs":{"formula_height":"1.593","formula_width":"1.717","image_id":"7313910086018220072","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/54f98a1dd8d8477d98438d235ec90555~tplv-xv4ileqgde-image.image","value":"R"}},{"type":"text","text":"上的单调增函数，且它的"},{"type":"text","text":"反函数"},{"type":"text","text":"就是"},{"type":"formula","node_id":"lqb307sw1ij5","attrs":{"formula_height":"5.016","formula_width":"12.431","image_id":"7313912141096992768","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6431b639d2b246efb85ec527b1d58d60~tplv-xv4ileqgde-image.image","value":"y=\\frac{x^{3}+12}{5}"}},{"type":"text","text":"，所以原"},{"type":"internal_link","node_id":"lra9xouychj","attrs":{"docId":"2293567924838015583","target":1,"title":"方程","doc_id":"2293567924838015583"},"children":[{"type":"text","text":"方程"}]},{"type":"text","text":"的求解问题可以转化为求函数"},{"type":"formula","node_id":"lqb307swib4","attrs":{"formula_height":"2.652","formula_width":"13.509","image_id":"7313927707903950863","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/663be80dd6a54ba2b6816b1038394225~tplv-xv4ileqgde-image.image","value":"y=\\sqrt[3]{5x-12}"}},{"type":"text","text":"与"},{"type":"formula","node_id":"lqb307sw18e4","attrs":{"formula_height":"5.01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a，b \\right)"}},{"type":"text","text":"内有定义，如果对于任意的"},{"type":"formula","node_id":"lp2uy14p1ley","attrs":{"formula_height":"2.262","formula_width":"15.782","image_id":"7302443046652674088","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/794b270df76f419bbe340ace42480913~tplv-xv4ileqgde-image.image","value":"x_{1}，x_{2}\\in(a，b)"}},{"type":"text","text":"，当"},{"type":"formula","node_id":"lp2uy14pnge","attrs":{"formula_height":"1.561","formula_width":"7.581","image_id":"7294182253972832295","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fad0bf73cdfa468e82410fb2b0949e5a~tplv-xv4ileqgde-image.image","value":"x_{1}f(x_{2})"}},{"type":"text","text":"，则称函数"},{"type":"formula","node_id":"lp2uy14pbz2","attrs":{"formula_height":"2.262","formula_width":"4.299","image_id":"7205568125696213050","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f58ab15fc13d445ea86696b722e6667c~tplv-xv4ileqgde-image.image","value":"f(x)"}},{"type":"text","text":"在"},{"type":"formula","node_id":"lp2uy14p14bf","attrs":{"formula_height":"2.262","formula_width":"6.19","image_id":"7302440481558954036","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/30add0ea29b34d7ba33cc31b328cddc1~tplv-xv4ileqgde-image.image","value":"\\left( a，b \\right)"}},{"type":"text","text":"内单调减少。单调增加和单调减少的函数，统称为单调函数，相应的区间称为函数的单调区间"},{"type":"reference","node_id":"lp2srlxhm8x","attrs":{"content":"lp2srlxhm8x","data_index":29,"index":29}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lp2uh89z23pt","attrs":{},"children":[{"type":"text","text":"反函数"},{"type":"text","text":"的单调性"},{"type":"reference","node_id":"loxs5hgw17ui","attrs":{"content":"loxs5hgw17ui","data_index":8,"index":8}},{"type":"text","text":"：设函数"},{"type":"formula","node_id":"lomq9j0kcr7","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239705494510632998","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2edc1192bea34b4293ade53c692bd2cb~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"是定义域"},{"type":"formula","node_id":"lox1ly9fsos","attrs":{"formula_height":"1.545","formula_width":"1.873","image_id":"7238927333405491257","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/9d32dba66f1548cba30dd62293bddd12~tplv-xv4ileqgde-image.image","value":"D"}},{"type":"text","text":"上的单调增（减）函数，那么此函数存在着反函数，且反函数也是单调增（减）函数"},{"type":"reference","node_id":"lomozmj210qb","attrs":{"content":"lomozmj210qb","data_index":30,"index":30}},{"type":"text","text":"。即互为反函数的两个函数具有相同的单调性"},{"type":"reference","node_id":"loo3pw78nvz","attrs":{"content":"loo3pw78nvz","data_index":19,"index":19}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lp2ujgebpo0","attrs":{},"children":[{"type":"text","text":"反函数单调性的证明：任取"},{"type":"formula","node_id":"lp2uy14q15mg","attrs":{"formula_height":"2.262","formula_width":"14.099","image_id":"7302446294759538740","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/06ed0baeea2b45e681fdc81c809f85e2~tplv-xv4ileqgde-image.image","value":"y_{1}，y_{2}\\in f(D)"}},{"type":"text","text":"，且"},{"type":"formula","node_id":"lp2uy14q12hd","attrs":{"formula_height":"1.686","formula_width":"7.21","image_id":"7302446294759587892","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fe7bdbc11b4244a1a7c1ae020986afc0~tplv-xv4ileqgde-image.image","value":"y_{1}x_{2}"}},{"type":"text","text":"，函数为单调递增函数，必有"},{"type":"formula","node_id":"lp2uy14q10nx","attrs":{"formula_height":"1.686","formula_width":"7.21","image_id":"7302447289572982824","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/46525075c81c46df813f8b22fd8dd443~tplv-xv4ileqgde-image.image","value":"y_{1}>y_{2}"}},{"type":"text","text":"，则当"},{"type":"formula","node_id":"lp2uy14q21li","attrs":{"formula_height":"1.686","formula_width":"7.21","image_id":"7302447289572982824","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/46525075c81c46df813f8b22fd8dd443~tplv-xv4ileqgde-image.image","value":"y_{1}>y_{2}"}},{"type":"text","text":"时，"},{"type":"formula","node_id":"lp2uy14qedr","attrs":{"formula_height":"1.561","formula_width":"7.581","image_id":"7302444688819355688","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d671767c098a4147a4f917cf85253563~tplv-xv4ileqgde-image.image","value":"x_{1}>x_{2}"}},{"type":"text","text":"，有"},{"type":"formula","node_id":"lp2uy14q1pnw","attrs":{"formula_height":"2.565","formula_width":"17.923","image_id":"7302453359397699599","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f83727dde8f940fca0f5cdd05fb31e1d~tplv-xv4ileqgde-image.image","value":"f^{-1}(y_{1})>f^{-1}(y_{2})"}},{"type":"reference","node_id":"lp2uxuaibbe","attrs":{"content":"lp2uxuaibbe","data_index":8,"index":8}},{"type":"text","text":"。"}]},{"type":"bullet_list","node_id":"lp2uey0r1jrx","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"loo2xn4t1gp5","attrs":{},"children":[{"type":"text","text":"奇偶性"}]}]}]},{"type":"paragraph","node_id":"lp2uf3k41apv","attrs":{},"children":[{"type":"text","text":"函数奇偶性"},{"type":"text","text":"的概念：设函数"},{"type":"formula","node_id":"lp2uy14q1roq","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7294181781958459455","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0855dc981a464429afc02c92b2e9cbf5~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的定义域"},{"type":"formula","node_id":"lp2uy14q1px6","attrs":{"formula_height":"1.545","formula_width":"1.873","image_id":"7238927333405491257","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/9d32dba66f1548cba30dd62293bddd12~tplv-xv4ileqgde-image.image","value":"D"}},{"type":"text","text":"关于原点对称，如果对任意的"},{"type":"formula","node_id":"lp2uy14q10mj","attrs":{"formula_height":"1.636","formula_width":"5.933","image_id":"7240072304753901624","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d8398687a218418b90daa8f29ea30bdf~tplv-xv4ileqgde-image.image","value":"x\\in D"}},{"type":"text","text":"，有"},{"type":"formula","node_id":"lp2uy14q1gvn","attrs":{"formula_height":"2.262","formula_width":"13.375","image_id":"7302447551897157672","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/14c137757343425ba18d07e1c958f62b~tplv-xv4ileqgde-image.image","value":"f(-x)=f(x)"}},{"type":"text","text":"，则称"},{"type":"formula","node_id":"lp2uy14qiji","attrs":{"formula_height":"2.262","formula_width":"4.299","image_id":"7205500085528952836","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d14863f90f9d47fea7965457bb94650c~tplv-xv4ileqgde-image.image","value":"f(x)"}},{"type":"text","text":"为偶函数；若有"},{"type":"formula","node_id":"lp2uy14q1zy3","attrs":{"formula_height":"2.262","formula_width":"15.135","image_id":"7240072500082638909","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/aaa6f84d22334bdea5ea23fde0066306~tplv-xv4ileqgde-image.image","value":"f(-x)=-f(x)"}},{"type":"text","text":"，则称"},{"type":"formula","node_id":"lp2uy14q4py","attrs":{"formula_height":"2.262","formula_width":"4.299","image_id":"7205500085528952836","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d14863f90f9d47fea7965457bb94650c~tplv-xv4ileqgde-image.image","value":"f(x)"}},{"type":"text","text":"为奇函数"},{"type":"reference","node_id":"lp2t1ua012ca","attrs":{"content":"lp2t1ua012ca","data_index":31,"index":31}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lp2uens91zqj","attrs":{},"children":[{"type":"text","text":"反函数"},{"type":"text","text":"的奇偶性"},{"type":"reference","node_id":"loxs5m3b1zew","attrs":{"content":"loxs5m3b1zew","data_index":8,"index":8}},{"type":"text","text":"：若奇函数存在反函数，则其反函数也是奇函数；定义域为非单元素集的偶函数不存在"},{"type":"text","text":"反函数"},{"type":"text","text":"，例如，函数"},{"type":"formula","node_id":"loo5krfdb4l","attrs":{"formula_height":"2.262","formula_width":"18.012","image_id":"7298629717127741440","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d40d490350894e95b12616f9c4f53aa4~tplv-xv4ileqgde-image.image","value":"f(x)=c，x\\in\\left\\{ 0 \\right\\}"}},{"type":"text","text":"，"},{"type":"formula","node_id":"loxrtstjdo4","attrs":{"formula_height":"1.025","formula_width":"0.98","image_id":"7264784666987593740","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/016ae9c1ca494694a18c9fdde2b09802~tplv-xv4ileqgde-image.image","value":"c"}},{"type":"text","text":"为"},{"type":"internal_link","node_id":"lpf6zly4cyx","attrs":{"docId":"315956147310414206","target":1,"title":"常数","doc_id":"315956147310414206"},"children":[{"type":"text","text":"常数"}]},{"type":"text","text":"，仅这样的偶函数存在反函数"},{"type":"reference","node_id":"loo3nseg17xp","attrs":{"content":"loo3nseg17xp","data_index":32,"index":32}},{"type":"text","text":"。"}]},{"type":"bullet_list","node_id":"lp2ufuat1mj8","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"loo3cny44sf","attrs":{},"children":[{"type":"text","text":"反函数"},{"type":"text","text":"的反函数"}]}]}]},{"type":"paragraph","node_id":"lp2ufoyx13c","attrs":{},"children":[{"type":"text","text":"反函数的反函数：若函数"},{"type":"formula","node_id":"loxrtstj1tlx","attrs":{"formula_height":"2.262","formula_width":"24.633","image_id":"7301152236711919670","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/efdd361d5b414e1ebd4241fca32e54c1~tplv-xv4ileqgde-image.image","value":"y=f(x)(x\\in D，y\\in M)"}},{"type":"text","text":"存在反函数"},{"type":"formula","node_id":"lqaqvwge10jv","attrs":{"formula_height":"2.565","formula_width":"26.985","image_id":"7313850765867794472","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fc47fb3433464a1380f5472969d649b2~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)(x\\in M，y\\in D)"}},{"type":"text","text":"，则"},{"type":"formula","node_id":"loohrb6d9wh","attrs":{"formula_height":"2.565","formula_width":"21.535","image_id":"7298725820816932898","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/1b1895c65d2d4c2cbafeb94e822aa0ff~tplv-xv4ileqgde-image.image","value":"f^{-1}\\left[ f(x) \\right]=x(x\\in D)"}},{"type":"text","text":"，"},{"type":"formula","node_id":"loxrtstj1l9p","attrs":{"formula_height":"2.791","formula_width":"22.668","image_id":"7301151297678491667","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/c71c396e6d8b4235a1cee16b27d86edc~tplv-xv4ileqgde-image.image","value":"f\\left[ f^{-1}(x) \\right]=x(x\\in M)"}},{"type":"text","text":"，即反函数的反函数就是原函数"},{"type":"reference","node_id":"lqaql43c1h7z","attrs":{"content":"lqaql43c1h7z","data_index":33,"index":33}},{"type":"reference","node_id":"loo3mykb1npv","attrs":{"content":"loo3mykb1npv","data_index":32,"index":32}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lqaqqpu65tz","attrs":{},"children":[{"type":"text","text":"例：已知"},{"type":"formula","node_id":"lqaqvwge178w","attrs":{"formula_height":"2.754","formula_width":"16.699","image_id":"7313857483028758563","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/58f5f1e71bd94e0989e9cec11ae9eef9~tplv-xv4ileqgde-image.image","value":"f(x)=4\\sqrt{2-3x}"}},{"type":"text","text":"，分别求出"},{"type":"formula","node_id":"lqaqvwge24vz","attrs":{"formula_height":"2.791","formula_width":"49.035","image_id":"7313855163972927540","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e73b0ff2523145cdab128da0339581d1~tplv-xv4ileqgde-image.image","value":"f^{-1}\\left[ f(1) \\right]，f\\left[ f^{-1}(1) \\right]，f^{-1}\\left[ f(-1) \\right]，f\\left[ f^{-1}(-1) \\right]"}},{"type":"text","text":"的值。"}]},{"type":"paragraph","node_id":"lqaqte9vmoi","attrs":{},"children":[{"type":"text","text":"解："},{"type":"formula","node_id":"lqb307sw24c3","attrs":{"formula_height":"2.262","formula_width":"4.299","image_id":"7241535197731094585","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/c9066f17e0814fbca51bc52410671bb5~tplv-xv4ileqgde-image.image","value":"f(x)"}},{"type":"text","text":"的定义域"},{"type":"formula","node_id":"lqb307sw1380","attrs":{"formula_height":"5.428","formula_width":"16.469","image_id":"7313899938017116200","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/04ed801e3a874bfeb7db963c9d772e7f~tplv-xv4ileqgde-image.image","value":"A=\\left\\{ x| x \\leq\\frac{2}{3} \\right\\}"}},{"type":"text","text":"，"},{"type":"formula","node_id":"lqb307sw1dtu","attrs":{"formula_height":"2.262","formula_width":"4.299","image_id":"7241535197731094585","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/c9066f17e0814fbca51bc52410671bb5~tplv-xv4ileqgde-image.image","value":"f(x)"}},{"type":"text","text":"的"},{"type":"internal_link","node_id":"lqj1re4k1sgb","attrs":{"docId":"6087720419503733950","target":1,"title":"值域","doc_id":"6087720419503733950"},"children":[{"type":"text","text":"值域"}]},{"type":"text","text":"，即"},{"type":"formula","node_id":"lqb307sw157t","attrs":{"formula_height":"2.565","formula_width":"6.651","image_id":"7313904547905290292","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/9847955ee57c4ac9bbd53f00ea88206b~tplv-xv4ileqgde-image.image","value":"f^{-1}(x)"}},{"type":"text","text":"的定义域"},{"type":"formula","node_id":"lqb307sw16yu","attrs":{"formula_height":"2.262","formula_width":"14.362","image_id":"7313899938017132584","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7fbdf16aa93643cfa636165c00f8019e~tplv-xv4ileqgde-image.image","value":"B=\\left\\{ x| x \\geq0\\right\\}"}},{"type":"text","text":"，由"},{"type":"text","text":"反函数"},{"type":"text","text":"的反函数性质，"},{"type":"formula","node_id":"lqb307sw1wo7","attrs":{"formula_height":"1.636","formula_width":"5.614","image_id":"7313904188898050100","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/9cc9c7c5d26f47759b12b646fa5225a7~tplv-xv4ileqgde-image.image","value":"1\\in B"}},{"type":"text","text":"，知"},{"type":"formula","node_id":"lqb307swmaf","attrs":{"formula_height":"2.791","formula_width":"14.145","image_id":"7313907984873127976","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/50f9188a59514f6c9a5db5561ee95c47~tplv-xv4ileqgde-image.image","value":"f\\left[ f^{-1}(1) \\right]=1"}},{"type":"text","text":"，而"},{"type":"formula","node_id":"lqb307sw118l","attrs":{"formula_height":"2.106","formula_width":"5.594","image_id":"7313908108487655459","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/c6459a7485da418eaebe522ea8b55408~tplv-xv4ileqgde-image.image","value":"1\\notin A"}},{"type":"text","text":"，知"},{"type":"formula","node_id":"lqb307sw1tab","attrs":{"formula_height":"2.565","formula_width":"9.367","image_id":"7313904547905323060","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f05edba895394fcf9f5569ff3f0e0a21~tplv-xv4ileqgde-image.image","value":"f^{-1}\\left[ f(1) \\right]"}},{"type":"text","text":"无意义"},{"type":"text","text":"，由"},{"type":"formula","node_id":"lqb307sw218z","attrs":{"formula_height":"1.805","formula_width":"7.354","image_id":"7313909190437814307","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8569790a4fdc436d8fddf11b3ba71dd2~tplv-xv4ileqgde-image.image","value":"-1\\in A"}},{"type":"text","text":"，"},{"type":"formula","node_id":"lqb307swovt","attrs":{"formula_height":"2.106","formula_width":"7.375","image_id":"7313904188898082868","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/2b705fc1c093435db5ae4c47a844a6ec~tplv-xv4ileqgde-image.image","value":"-1\\notin B"}},{"type":"text","text":"，可知"},{"type":"formula","node_id":"lqb307sw24ee","attrs":{"formula_height":"2.565","formula_width":"17.036","image_id":"7313910086018023464","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5f6c6109bc50492a8a078ab3b5689264~tplv-xv4ileqgde-image.image","value":"f^{-1}\\left[ f(-1) \\right]=-1"}},{"type":"text","text":"，"},{"type":"formula","node_id":"lqb307sw1l2s","attrs":{"formula_height":"2.791","formula_width":"11.756","image_id":"7313903054238236706","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0c71c7b46dcf445d8f8a6ece346da76a~tplv-xv4ileqgde-image.image","value":"f\\left[ f^{-1} (-1)\\right]"}},{"type":"text","text":"无意义"},{"type":"reference","node_id":"lqazjsr917fe","attrs":{"content":"lqazjsr917fe","data_index":33,"index":33}},{"type":"text","text":"。"}]},{"type":"heading","node_id":"lp2d4u241fxw","attrs":{"level":1},"children":[{"type":"text","text":"一些函数的反函数"}]},{"type":"heading","node_id":"lowo2djsbph","attrs":{"level":2},"children":[{"type":"text","text":"分段函数的反函数"}]},{"type":"bullet_list","node_id":"lp2luzqd220o","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lp2luilryhj","attrs":{},"children":[{"type":"text","text":"分段函数的概念"}]}]}]},{"type":"paragraph","node_id":"lp2lv1qk1cm","attrs":{},"children":[{"type":"text","text":"若一元函数"},{"type":"formula","node_id":"lp2s5suo5es","attrs":{"formula_height":"2.262","formula_width":"4.299","image_id":"7229323141510594599","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ffabcba2e09e49c5972dc5779b4f4753~tplv-xv4ileqgde-image.image","value":"f(x)"}},{"type":"text","text":"的定义域由若干段组成（每段可以几个区间的并或一个区间或一个点），在不同段上有相应的表达式，则称在这个定义域上定义了一个分段函数。这种函数的对应关系是分段表示的，称为分段定义的函数，简称分段函数"},{"type":"reference","node_id":"lp2q6n5rv51","attrs":{"content":"lp2q6n5rv51","data_index":34,"index":34}},{"type":"text","text":"。"}]},{"type":"bullet_list","node_id":"lowokbg3ly4","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"loy32a1ci9s","attrs":{},"children":[{"type":"internal_link","node_id":"lra9xouvz4s","attrs":{"docId":"8607972194790743201","target":1,"title":"定理","doc_id":"8607972194790743201"},"children":[{"type":"text","text":"定理"}]}]}]}]},{"type":"paragraph","node_id":"lowonl3m3ut","attrs":{},"children":[{"type":"text","text":"已知函数"},{"type":"formula","node_id":"lowp3bnbit3","attrs":{"formula_height":"11.765","formula_width":"23.704","image_id":"7300871536859234367","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ea22ad44b6f842a7a437cef1f4199c6e~tplv-xv4ileqgde-image.image","value":"f(x)=\\left\\{\\begin{array}{l} \ng_{1}(x)，x\\in L_{1}\n\\newline\ng_{2}(x)，x\\in L_{2}\n\\newline\n……，……\n\\newline\ng_{n}(x)，x\\in L_{n}\n\\end{array}\\right."}},{"type":"text","text":"（其中"},{"type":"formula","node_id":"lpf6ysyv1c9f","attrs":{"formula_height":"2.363","formula_width":"18.54","image_id":"7305567382419062825","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/23f092865f094939a3497530cdcafcfc~tplv-xv4ileqgde-image.image","value":"L_{i}\\cap L_{j}=\\phi，i\\ne j"}},{"type":"text","text":"）。若函数"},{"type":"formula","node_id":"lowp3bnb1cov","attrs":{"formula_height":"2.262","formula_width":"30.666","image_id":"7300872033901019199","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ce4cfe8b859e4bd980e7046a47ae0466~tplv-xv4ileqgde-image.image","value":"g_{i}(x)(i=1，2，3，……，n)"}},{"type":"text","text":"在"},{"type":"formula","node_id":"lowp3bnb7ml","attrs":{"formula_height":"1.902","formula_width":"2.28","image_id":"7300872033901199423","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/a978a4e2b6af406e884d160773509d9d~tplv-xv4ileqgde-image.image","value":"L_{i}"}},{"type":"text","text":"上均存在"},{"type":"text","text":"反函数"},{"type":"formula","node_id":"lowp3bnbe32","attrs":{"formula_height":"2.679","formula_width":"6.366","image_id":"7300872272938827839","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/25f666aab37640ceb089cbd43de56d17~tplv-xv4ileqgde-image.image","value":"g_{i}^{-1}(x)"}},{"type":"text","text":"，且"},{"type":"formula","node_id":"lowp3bnb1t32","attrs":{"formula_height":"2.363","formula_width":"25.303","image_id":"7300872569090015286","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/73bceb3810d54cf4a51de86ff969100b~tplv-xv4ileqgde-image.image","value":"g_{i}(L_{i})\\cap g_{j}(L_{j})=\\phi (i\\ne j)"}},{"type":"text","text":"，那么分段函数"},{"type":"formula","node_id":"lowp3bnbeue","attrs":{"formula_height":"2.262","formula_width":"4.299","image_id":"7205500085528952836","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d14863f90f9d47fea7965457bb94650c~tplv-xv4ileqgde-image.image","value":"f(x)\n"}},{"type":"text","text":"存在反函数，且反函数为"},{"type":"formula","node_id":"lowp3bnb8a8","attrs":{"formula_height":"12.779","formula_width":"31.043","image_id":"7300873554503974924","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/98ca5a04a4034f97bd159e45a0f20992~tplv-xv4ileqgde-image.image","value":"f^{-1}(x)=\\left\\{\\begin{array}{l} \ng_{1}^{-1}(x)，x\\in g_{1}(L_{1})\n\\newline\ng_{2}^{-1}(x)，x\\in g_{2}(L_{2})\n\\newline\n……，……\n\\newline\ng_{n}^{-1}(x)，x\\in g_{n}(L_{n})\n\\end{array}\\right."}},{"type":"reference","node_id":"lowp38hcwnj","attrs":{"content":"lowp38hcwnj","data_index":35,"index":35}},{"type":"text","text":"。"}]},{"type":"bullet_list","node_id":"loy32a1d1xpx","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lp2lvgec1f8u","attrs":{},"children":[{"type":"text","text":"实例"}]}]}]},{"type":"paragraph","node_id":"lp2lvr3m1hwq","attrs":{},"children":[{"type":"text","text":"判断函数"},{"type":"formula","node_id":"lp2s5suo1i6b","attrs":{"formula_height":"6.785","formula_width":"28.302","image_id":"7302420697249267752","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/dea84afd8db64ad48931ed9cfed0639e~tplv-xv4ileqgde-image.image","value":"f(x)=\\left\\{\\begin{array}{l} \n3x^{2}+1，0\\beta"}},{"type":"text","text":"时的"},{"type":"formula","node_id":"lpf6ysyvaow","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7305624323115548691","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/9cf72ffba73842dfa17ab5b9bd41324a~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"的一切值，有"},{"type":"formula","node_id":"lpf6ysyvp67","attrs":{"formula_height":"2.262","formula_width":"14.09","image_id":"7305613825041170443","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ec9be55651d94b19811b7f07a9b6fce1~tplv-xv4ileqgde-image.image","value":"\\left| f(x )-A\\right|<\\varepsilon"}},{"type":"text","text":"，那么，数"},{"type":"formula","node_id":"lpf6ysyv8xm","attrs":{"formula_height":"1.62","formula_width":"1.697","image_id":"7205507815828357178","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/75d86a10adb14faaa296f0f78f00eefb~tplv-xv4ileqgde-image.image","value":"A"}},{"type":"text","text":"叫做函数"},{"type":"formula","node_id":"lpf6ysyv5ug","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的极限。记作"},{"type":"formula","node_id":"lpf6ysyv1jyq","attrs":{"formula_height":"3.345","formula_width":"13.194","image_id":"7305629793947500555","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/c628f0661e7d45069828b3dad1c76fee~tplv-xv4ileqgde-image.image","value":"\\lim_{n >\\infty}{f(x)}=A"}},{"type":"text","text":"或"},{"type":"formula","node_id":"lpf6ysyw170j","attrs":{"formula_height":"2.262","formula_width":"9.515","image_id":"7305614074921025575","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/e40d8d472aaf412cb95d1cc5a6800eec~tplv-xv4ileqgde-image.image","value":"f(x)\\rightarrow A"}},{"type":"text","text":"（当"},{"type":"formula","node_id":"lpf6ysywnwt","attrs":{"formula_height":"1.181","formula_width":"7.076","image_id":"7305614074921041959","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/516824c632304bceb5143e2ea4aca748~tplv-xv4ileqgde-image.image","value":"x\\rightarrow \\infty"}},{"type":"text","text":"时）。因为函数"},{"type":"formula","node_id":"lpf6ysywjsa","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的自变量"},{"type":"formula","node_id":"lpf6ysywnq9","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7205530821682266123","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5d6a0958b3bb49afa5681fa8402e6f79~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"是取实数值，即是函数"},{"type":"formula","node_id":"lpf6ysyw1a0x","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的定义域在区间"},{"type":"formula","node_id":"lpf6ysywr8n","attrs":{"formula_height":"2.262","formula_width":"13.073","image_id":"7305615369904029715","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/670c0554ceeb46e0b57e02696c16289f~tplv-xv4ileqgde-image.image","value":"(-\\infty，+\\infty)"}},{"type":"text","text":"内，所以当"},{"type":"formula","node_id":"lpf6ysywi1c","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7205530821682266123","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5d6a0958b3bb49afa5681fa8402e6f79~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"都取正值，就可记作"},{"type":"formula","node_id":"lpf6ysywwby","attrs":{"formula_height":"3.412","formula_width":"14.794","image_id":"7305629793947533323","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6692f43481bd48a0a174fdf654aaa00e~tplv-xv4ileqgde-image.image","value":"\\lim_{n \\rightarrow+\\infty}{f(x)}=A"}},{"type":"text","text":"；当"},{"type":"formula","node_id":"lpf6ysywaem","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7205530821682266123","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5d6a0958b3bb49afa5681fa8402e6f79~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"都取负值，便记作"},{"type":"formula","node_id":"lpf6ysywd48","attrs":{"formula_height":"3.412","formula_width":"14.794","image_id":"7305620233576955940","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d95e50fd8a14474b9dd4a7aea7fc2303~tplv-xv4ileqgde-image.image","value":"\\lim_{n \\rightarrow-\\infty}{f(x)}=A"}},{"type":"reference","node_id":"lpf31as61aqj","attrs":{"content":"lpf31as61aqj","data_index":39,"index":39}},{"type":"text","text":"。"}]},{"type":"bullet_list","node_id":"lp2mwenh1m8h","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lp2mw2ah1d0h","attrs":{},"children":[{"type":"internal_link","node_id":"lqj1re4kyn7","attrs":{"docId":"385229163457989516","target":1,"title":"导数","doc_id":"385229163457989516"},"children":[{"type":"text","text":"导数"}]},{"type":"text","text":"概念"}]}]}]},{"type":"paragraph","node_id":"lp2mwcajzku","attrs":{},"children":[{"type":"text","text":"设函数"},{"type":"formula","node_id":"lp2s5suo21ki","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lp2s5suo159g","attrs":{"formula_height":"1.375","formula_width":"2.282","image_id":"7245605966522089475","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/91216d8b2d7e4cada968e4b8fc452e69~tplv-xv4ileqgde-image.image","value":"x_{0}"}},{"type":"text","text":"的某"},{"type":"text","text":"邻域"},{"type":"text","text":"内有定义，当自变量"},{"type":"formula","node_id":"lp2s5suoc1b","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7205530821682266123","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5d6a0958b3bb49afa5681fa8402e6f79~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"在"},{"type":"formula","node_id":"lp2s5suo11vu","attrs":{"formula_height":"1.375","formula_width":"2.282","image_id":"7245605966522089475","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/91216d8b2d7e4cada968e4b8fc452e69~tplv-xv4ileqgde-image.image","value":"x_{0}"}},{"type":"text","text":"处取得增量"},{"type":"formula","node_id":"lp2s5suo1ba9","attrs":{"formula_height":"2.262","formula_width":"21.583","image_id":"7302401895438548992","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/4ba53aeb7d354304ad5434375cfa8b5d~tplv-xv4ileqgde-image.image","value":"\\Delta x\\left[ x_{0} +\\Delta x\\in U(x_{0})\\right]"}},{"type":"text","text":"时，函数有相应的改变量"},{"type":"formula","node_id":"lp2s5suo1waw","attrs":{"formula_height":"2.262","formula_width":"25.293","image_id":"7296029018925711372","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/7a06f306506c4fe2847f6b52f2e66230~tplv-xv4ileqgde-image.image","value":"\\Delta y=f(x_{0}+\\Delta x)-f(x_{0})"}},{"type":"text","text":"。如果极限"},{"type":"formula","node_id":"lp2s5suo38i","attrs":{"formula_height":"5.088","formula_width":"37.518","image_id":"7302400073261137935","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d503a8a354174d108264415659fc319b~tplv-xv4ileqgde-image.image","value":"\\lim_{\\Delta x \\rightarrow 0}{\\frac{\\Delta y}{\\Delta x}}=\\lim_{\\Delta x \\rightarrow 0}{\\frac{f(x_{0}+\\Delta x)-f(x_{0})}{\\Delta x}}"}},{"type":"text","text":"存在，则称函数"},{"type":"formula","node_id":"lp2s5suo1y05","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lp2s5suo1cnd","attrs":{"formula_height":"1.375","formula_width":"2.282","image_id":"7245605966522089475","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/91216d8b2d7e4cada968e4b8fc452e69~tplv-xv4ileqgde-image.image","value":"x_{0}"}},{"type":"text","text":"处可导，并称此极限值为函数"},{"type":"formula","node_id":"lp2s5suor9a","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lp2s5suo1d0q","attrs":{"formula_height":"1.375","formula_width":"2.282","image_id":"7245605966522089475","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/91216d8b2d7e4cada968e4b8fc452e69~tplv-xv4ileqgde-image.image","value":"x_{0}"}},{"type":"text","text":"处的"},{"type":"internal_link","node_id":"lqj1re4z1eo4","attrs":{"docId":"385229163457989516","target":1,"title":"导数","doc_id":"385229163457989516"},"children":[{"type":"text","text":"导数"}]},{"type":"text","text":"，记作"},{"type":"formula","node_id":"lp2s5suoo9j","attrs":{"formula_height":"2.396","formula_width":"6.034","image_id":"7296039272769912843","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b2ef5caf74fc45798f6d0ca8c7fa46a8~tplv-xv4ileqgde-image.image","value":"f'(x_{0})"}},{"type":"text","text":"，即"},{"type":"formula","node_id":"lp2s5suo1mwl","attrs":{"formula_height":"5.088","formula_width":"46.57","image_id":"7302403139540516879","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5eeeaa6d078849c29765d1b8acf7db9f~tplv-xv4ileqgde-image.image","value":"f'(x_{0})=\\lim_{\\Delta x \\rightarrow 0}{\\frac{\\Delta y}{\\Delta x}}=\\lim_{\\Delta x \\rightarrow 0}{\\frac{f(x_{0}+\\Delta x)-f(x_{0})}{\\Delta x}}"}},{"type":"reference","node_id":"lp2n3l0bq4n","attrs":{"content":"lp2n3l0bq4n","data_index":40,"index":40}},{"type":"text","text":"。"}]},{"type":"bullet_list","node_id":"lp2d7zcn1ksu","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lp2d7zcn5yg","attrs":{},"children":[{"type":"text","text":"求导法则"}]}]}]},{"type":"paragraph","node_id":"lowia0lqq25","attrs":{},"children":[{"type":"text","text":"若函数"},{"type":"formula","node_id":"loohrb6d1f60","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lowo3hnbu5z","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7204367456167526433","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8a001b31906c4e76943a171550d8f84a~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"的某"},{"type":"text","text":"邻域"},{"type":"text","text":"内连续，严格单调，且"},{"type":"formula","node_id":"lp2s5suohwd","attrs":{"formula_height":"2.396","formula_width":"9.075","image_id":"7302403139540402191","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/20834ab40aff4c3491d1712ea96e36ff~tplv-xv4ileqgde-image.image","value":"f{'}(x)\\ne0"}},{"type":"text","text":"，则它"},{"type":"text","text":"反函数"},{"type":"formula","node_id":"lowo3hnb1zio","attrs":{"formula_height":"2.565","formula_width":"18.142","image_id":"7300860696567463975","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/9bff3c7b05f546f6b6021dd63be3a0b7~tplv-xv4ileqgde-image.image","value":"x=f^{-1}(y)=\\varphi(y)"}},{"type":"text","text":"在"},{"type":"formula","node_id":"lowo3hnb1t28","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205526198313549862","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fdcdd22bc657443e9666202d8c51ebd6~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"处可导，且"},{"type":"formula","node_id":"loxrtstj1jnr","attrs":{"formula_height":"5.208","formula_width":"26.218","image_id":"7300861027422584886","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/47346f03bb3c499ca7f7815ca33e3378~tplv-xv4ileqgde-image.image","value":"\\left[ f^{-1} (x)\\right]'=\\varphi'(y)=\\frac{1}{f'(x)}"}},{"type":"reference","node_id":"loohmmnj1f6x","attrs":{"content":"loohmmnj1f6x","data_index":41,"index":41}},{"type":"reference","node_id":"lowjkikiobm","attrs":{"content":"lowjkikiobm","data_index":13,"index":13}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lp2d81n31vi3","attrs":{},"children":[{"type":"text","text":"证明：函数"},{"type":"formula","node_id":"lowo3hnb20em","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"存在反函数"},{"type":"formula","node_id":"lowo3hnb1alj","attrs":{"formula_height":"2.565","formula_width":"18.142","image_id":"7300860832777355305","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5569683c3c9f4e0b9c68b477466c5783~tplv-xv4ileqgde-image.image","value":"x=f^{-1}(y)=\\varphi(y)"}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lowbzxm51xuv","attrs":{},"children":[{"type":"text","text":"设"},{"type":"formula","node_id":"lowo3hnb24yv","attrs":{"formula_height":"2.262","formula_width":"8.66","image_id":"7300860772174020627","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/f50c0799f4ae47bb90ebb37c8ad1ef4f~tplv-xv4ileqgde-image.image","value":"x=\\varphi(y)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lowo3hnb1xsh","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7195924942494367756","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ce156f8b40094293952ca9474150dc9b~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"的自变量的改变量是"},{"type":"formula","node_id":"lowo3hnb1l5c","attrs":{"formula_height":"2.262","formula_width":"11.895","image_id":"7300814943841157139","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/97b0e9690eac42e283822d79e11e138b~tplv-xv4ileqgde-image.image","value":"\\Delta y(\\Delta y\\ne0)"}},{"type":"text","text":"，则"},{"type":"formula","node_id":"lowo3hnb5qg","attrs":{"formula_height":"2.262","formula_width":"23.417","image_id":"7300860914797002793","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/71cafe36324644009e69253a72006d1d~tplv-xv4ileqgde-image.image","value":"\\Delta x=\\varphi(y+\\Delta y)-\\varphi(y)"}},{"type":"text","text":"，"},{"type":"formula","node_id":"lpf6ysywwwy","attrs":{"formula_height":"2.262","formula_width":"23.318","image_id":"7305571261483286547","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/98724d6d618349ff83ac7786cf9860f3~tplv-xv4ileqgde-image.image","value":"\\Delta y=f(x+\\Delta x)-f(x)"}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lowiuby5g4t","attrs":{},"children":[{"type":"text","text":"因函数"},{"type":"formula","node_id":"lowo3hnb1lh9","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lowo3hnbtqs","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7204367456167526433","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8a001b31906c4e76943a171550d8f84a~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"的某"},{"type":"text","text":"邻域"},{"type":"text","text":"内是严格单调的"},{"type":"internal_link","node_id":"lqj1re4kg6g","attrs":{"docId":"6119487096335615130","target":1,"title":"连续函数","doc_id":"6119487096335615130"},"children":[{"type":"text","text":"连续函数"}]},{"type":"text","text":"，所以"},{"type":"text","text":"反函数"},{"type":"formula","node_id":"lowo3hnb23ze","attrs":{"formula_height":"2.565","formula_width":"18.142","image_id":"7300861041657823251","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/09b166f4b5204252822b34bf125916f8~tplv-xv4ileqgde-image.image","value":"x=f^{-1}(y)=\\varphi(y)"}},{"type":"text","text":"在"},{"type":"formula","node_id":"lowo3hnbq4q","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7195924942494367756","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ce156f8b40094293952ca9474150dc9b~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"的某邻域内也是严格单调的连续函数，因此当"},{"type":"formula","node_id":"lowo3hnb1qyd","attrs":{"formula_height":"2.084","formula_width":"7.644","image_id":"7300819618996879372","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/013509c68df342d8a3a332c889577508~tplv-xv4ileqgde-image.image","value":"\\Delta y\\rightarrow0"}},{"type":"text","text":"时有"},{"type":"formula","node_id":"lowo3hnbi9n","attrs":{"formula_height":"1.67","formula_width":"7.829","image_id":"7297910360055693323","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5ec31a74f0a0481884c62fa3761041c0~tplv-xv4ileqgde-image.image","value":"\\Delta x\\rightarrow0"}},{"type":"text","text":"；当"},{"type":"formula","node_id":"lowo3hnb1zk2","attrs":{"formula_height":"2.106","formula_width":"7.142","image_id":"7300820668034138151","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/63e32645de404710b05ca279690dfa4a~tplv-xv4ileqgde-image.image","value":"\\Delta y\\ne0"}},{"type":"text","text":"时，有"},{"type":"formula","node_id":"lowo3hnb7ro","attrs":{"formula_height":"2.106","formula_width":"7.327","image_id":"7249698773562032128","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/47d63d2efd174f5ca1e214091e1e921a~tplv-xv4ileqgde-image.image","value":"\\Delta x\\ne0"}},{"type":"text","text":"，于是，"},{"type":"formula","node_id":"lowo3hnb1qd1","attrs":{"formula_height":"6.103","formula_width":"11.43","image_id":"7300823961187647551","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/383e22a06cc34a31bc31aba8e3d4139f~tplv-xv4ileqgde-image.image","value":"\\frac{\\Delta x}{\\Delta y}=\\frac{1}{\\frac{\\Delta y}{\\Delta x}}"}},{"type":"text","text":"，由此可得"},{"type":"formula","node_id":"lqacalyv23u4","attrs":{"formula_height":"6.103","formula_width":"44.885","image_id":"7313750712557797391","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6d2e14fc57b34e5bb4547d761721f1c6~tplv-xv4ileqgde-image.image","value":"\\lim_{\\Delta y \\rightarrow 0}{\\frac{\\Delta x}{\\Delta y}}=\\lim_{\\Delta x \\rightarrow 0}{\\frac{1}{\\frac{\\Delta y}{\\Delta x}}}=1/{\\lim_{\\Delta x\\rightarrow 0}{\\frac{\\Delta y}{\\Delta x}}}=\\frac{1}{f'(x)}"}},{"type":"text","text":"，即"},{"type":"text","text":"反函数"},{"type":"formula","node_id":"lowo3hnb24m6","attrs":{"formula_height":"2.565","formula_width":"18.142","image_id":"7300861041657921555","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/026b8811064c4143bd28c4b4a15f0458~tplv-xv4ileqgde-image.image","value":"x=f^{-1}(y)=\\varphi(y)"}},{"type":"text","text":"在"},{"type":"formula","node_id":"lowo3hnb1ga1","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7195924942494367756","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ce156f8b40094293952ca9474150dc9b~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"处可导，且"},{"type":"formula","node_id":"lowo3hnbm68","attrs":{"formula_height":"5.208","formula_width":"26.218","image_id":"7300861027422584886","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/47346f03bb3c499ca7f7815ca33e3378~tplv-xv4ileqgde-image.image","value":"\\left[ f^{-1} (x)\\right]'=\\varphi'(y)=\\frac{1}{f'(x)}"}},{"type":"text","text":"，即"},{"type":"text","text":"反函数"},{"type":"text","text":"的"},{"type":"internal_link","node_id":"lra9xouz1nqg","attrs":{"docId":"385229163457989516","target":1,"title":"导数","doc_id":"385229163457989516"},"children":[{"type":"text","text":"导数"}]},{"type":"text","text":"等于直接函数导数的"},{"type":"internal_link","node_id":"lqj1re4l1hqg","attrs":{"docId":"1935161021436582380","target":1,"title":"倒数","doc_id":"1935161021436582380"},"children":[{"type":"text","text":"倒数"}]},{"type":"reference","node_id":"lowjjbugsx6","attrs":{"content":"lowjjbugsx6","data_index":13,"index":13}},{"type":"text","text":"。"}]},{"type":"bullet_list","node_id":"lqb11r3cs42","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lqanpyj8whg","attrs":{},"children":[{"type":"text","text":"多元函数"},{"type":"internal_link","node_id":"lqbc5z6m1shi","attrs":{"docId":"7251181273717487029","target":1,"title":"偏导数","doc_id":"7251181273717487029"},"children":[{"type":"text","text":"偏导数"}]},{"type":"text","text":"的概念"}]}]}]},{"type":"paragraph","node_id":"lqb11fq91yzj","attrs":{},"children":[{"type":"text","text":"设"},{"type":"formula","node_id":"lqb307sx1oca","attrs":{"formula_height":"2.262","formula_width":"35.057","image_id":"7313915822450294824","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0401b82a47ab4ccdbada3c4025f3c504~tplv-xv4ileqgde-image.image","value":"y=f(x)=f(x_{1}，……，x_{m})\\in R"}},{"type":"text","text":"，在"},{"type":"formula","node_id":"lqb307sx9g1","attrs":{"formula_height":"1.025","formula_width":"1.986","image_id":"7313931869911777295","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/b2de9c682fc04ca5a5b48a5a5029d035~tplv-xv4ileqgde-image.image","value":"m"}},{"type":"text","text":"个自变量中，固定"},{"type":"formula","node_id":"lqb307sx1a5w","attrs":{"formula_height":"1.692","formula_width":"5.883","image_id":"7313915087243460642","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/44188f2b8ec247cbaab6bf49c4a6e558~tplv-xv4ileqgde-image.image","value":"m-1"}},{"type":"text","text":"个："},{"type":"formula","node_id":"lqb307sx1gz1","attrs":{"formula_height":"2.167","formula_width":"37.506","image_id":"7313916861228449844","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/9e1b977560654ffdb514e8ace837f74a~tplv-xv4ileqgde-image.image","value":"x_{1}，……，x_{i-1}，x_{i+1}，……，x_{m}"}},{"type":"text","text":"，让"},{"type":"formula","node_id":"lqb307sx20hy","attrs":{"formula_height":"1.357","formula_width":"2.034","image_id":"7313924260001579043","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fcd3af163b654e669ec0059823366afb~tplv-xv4ileqgde-image.image","value":"x_{i}"}},{"type":"text","text":"变化，那么可视"},{"type":"formula","node_id":"lqb307sx23rw","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7313926136059199528","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ffdf5a3f89b84a5d9603d5343c7211ab~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"仅仅是"},{"type":"formula","node_id":"lqb307sxlao","attrs":{"formula_height":"1.357","formula_width":"2.034","image_id":"7313924260001579043","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fcd3af163b654e669ec0059823366afb~tplv-xv4ileqgde-image.image","value":"x_{i}"}},{"type":"text","text":"的一元函数，"},{"type":"formula","node_id":"lqb307sx1sve","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7313924846759493684","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/48d4bdc6cdac4cec8533471eb475dae4~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"对"},{"type":"formula","node_id":"lqb307sxft6","attrs":{"formula_height":"1.357","formula_width":"2.034","image_id":"7313924260001579043","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fcd3af163b654e669ec0059823366afb~tplv-xv4ileqgde-image.image","value":"x_{i}"}},{"type":"text","text":"求导数，便是多元函数的偏"},{"type":"internal_link","node_id":"m0bjiw7kl74","attrs":{"docId":"385229163457989516","target":1,"title":"导数","doc_id":"385229163457989516"},"children":[{"type":"text","text":"导数"}]},{"type":"reference","node_id":"lqb204vb1er2","attrs":{"content":"lqb204vb1er2","data_index":42,"index":42}},{"type":"text","text":"。"}]},{"type":"bullet_list","node_id":"lqb11srb1th6","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lqanqfig1xvq","attrs":{},"children":[{"type":"text","text":"多元函数的反函数求导"},{"type":"internal_link","node_id":"lra9xouxulw","attrs":{"docId":"8607972194790743201","target":1,"title":"定理","doc_id":"8607972194790743201"},"children":[{"type":"text","text":"定理"}]}]}]}]},{"type":"paragraph","node_id":"lqanqbgo59k","attrs":{},"children":[{"type":"text","text":"设可微函数"},{"type":"formula","node_id":"lqbppk6l14bq","attrs":{"formula_height":"2.262","formula_width":"11.739","image_id":"7314106209861124111","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/1b542f20fbce470f86bbeb5fd3256ac3~tplv-xv4ileqgde-image.image","value":"z=f(x，y)"}},{"type":"text","text":"存在两个可微的分别偏"},{"type":"formula","node_id":"lqbppk6l795","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205531495279099938","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/67b74808db244ad19e73380f2fd4f484~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"和偏"},{"type":"formula","node_id":"lqbppk6lpw3","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7205535490290745382","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/d7b08777d0f743ab852bb716b756882e~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"的反函数"},{"type":"formula","node_id":"lqbppk6l1kn8","attrs":{"formula_height":"2.262","formula_width":"11.574","image_id":"7314012558015512576","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/4847b2e66653436aa24bd268cc7608d4~tplv-xv4ileqgde-image.image","value":"y=g(x，z)"}},{"type":"text","text":"和"},{"type":"formula","node_id":"lqbppk6l1ryq","attrs":{"formula_height":"2.262","formula_width":"11.798","image_id":"7314012558015528960","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/0d786685eb78482f96bf1fc2f95e1451~tplv-xv4ileqgde-image.image","value":"x=h(y，z)"}},{"type":"text","text":"，且这两个函数在相应定义区域"},{"type":"formula","node_id":"lqbppk6lcfw","attrs":{"formula_height":"1.902","formula_width":"3.72","image_id":"7314114591699501108","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/fbb42ea3832845199a3b437a325954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y}{\\partial z}\\ne 0，\\frac{\\partial x}{\\partial z}\\ne 0"}},{"type":"text","text":"，则可微函数"},{"type":"formula","node_id":"lqbppk6lis4","attrs":{"formula_height":"2.262","formula_width":"11.739","image_id":"7314106209861124111","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/1b542f20fbce470f86bbeb5fd3256ac3~tplv-xv4ileqgde-image.image","value":"z=f(x，y)"}},{"type":"text","text":"在相应定义区间"},{"type":"formula","node_id":"lqbppk6l7kt","attrs":{"formula_height":"2.213","formula_width":"3.76","image_id":"7314101957197611048","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/70abdfb098c249d6a3e023e57450d036~tplv-xv4ileqgde-image.image","value":"D_{xy}"}},{"type":"text","text":"内两个"},{"type":"internal_link","node_id":"lqbpr0dx1lay","attrs":{"docId":"7251181273717487029","target":1,"title":"偏导数","doc_id":"7251181273717487029"},"children":[{"type":"text","text":"偏导数"}]},{"type":"formula","node_id":"lqbppk6l6v8","attrs":{"formula_height":"5.163","formula_width":"9.217","image_id":"7314106777740705807","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/43e033ce2e4848338ce51c8594b84b4e~tplv-xv4ileqgde-image.image","value":"\\frac{\\partial z}{\\partial y}、\\frac{\\partial z}{\\partial x}"}},{"type":"text","text":"存在，且"},{"type":"formula","node_id":"lqbppk6l1ddr","attrs":{"formula_height":"6.115","formula_width":"22.874","image_id":"7314012558015594496","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/45fabbae9c4d47b79302bf8b821e9644~tplv-xv4ileqgde-image.image","value":"\\frac{\\partial z}{\\partial y}=\\frac{1}{\\frac{\\partial y}{\\partial x}}，\\frac{\\partial z}{\\partial x}=\\frac{1}{\\frac{\\partial x}{\\partial z}}"}},{"type":"reference","node_id":"lqbpgf0xdms","attrs":{"content":"lqbpgf0xdms","data_index":43,"index":43}},{"type":"text","text":"。"}]},{"type":"bullet_list","node_id":"lp2d842eqtt","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lp2d842e1nkl","attrs":{},"children":[{"type":"text","text":"几何意义"}]}]}]},{"type":"paragraph","node_id":"lowd3yiq2534","attrs":{},"children":[{"type":"text","text":"已知函数"},{"type":"formula","node_id":"lowo3hnb1ycp","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lowo3hnb4ta","attrs":{"formula_height":"1.025","formula_width":"1.294","image_id":"7205530821682266123","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/5d6a0958b3bb49afa5681fa8402e6f79~tplv-xv4ileqgde-image.image","value":"x"}},{"type":"text","text":"处的"},{"type":"internal_link","node_id":"lqj1re4l1n94","attrs":{"docId":"385229163457989516","target":1,"title":"导数","doc_id":"385229163457989516"},"children":[{"type":"text","text":"导数"}]},{"type":"text","text":"，其几何意义是：曲线"},{"type":"formula","node_id":"lowo3hnb1y76","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lowo3hnb1vur","attrs":{"formula_height":"2.262","formula_width":"6.425","image_id":"7300777920556056617","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/26508c44090549dab426dae7b66a269c~tplv-xv4ileqgde-image.image","value":"(x，y)"}},{"type":"text","text":"处的"},{"type":"text","text":"切线"},{"type":"text","text":"斜率"},{"type":"text","text":"，如下图所示，则"},{"type":"formula","node_id":"loxrtstj1fgt","attrs":{"formula_height":"4.676","formula_width":"19.246","image_id":"7301129649617338379","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/6797a97e8e314e45a2a979e3a1f5086f~tplv-xv4ileqgde-image.image","value":"\\frac{dy}{dx}=f{'}(x)=tan\\alpha"}},{"type":"text","text":"，其中"},{"type":"formula","node_id":"lowo3hnb133l","attrs":{"formula_height":"1.025","formula_width":"1.448","image_id":"7278205819731050531","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/02c59f15e61f4114905803c16b1c4c42~tplv-xv4ileqgde-image.image","value":"\\alpha"}},{"type":"text","text":"为切线的倾斜角"},{"type":"reference","node_id":"lowhqzum24fc","attrs":{"content":"lowhqzum24fc","data_index":44,"index":44}},{"type":"text","text":"。"}]},{"type":"paragraph","node_id":"lowd8djhbd3","attrs":{},"children":[{"type":"text","text":"同理函数"},{"type":"formula","node_id":"lowo3hnb1tqq","attrs":{"formula_height":"2.262","formula_width":"8.424","image_id":"7239706616717639741","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/3a56d1af936f4381ae78b8fed9b6f3df~tplv-xv4ileqgde-image.image","value":"y=f(x)"}},{"type":"text","text":"的"},{"type":"text","text":"反函数"},{"type":"formula","node_id":"lowo3hnbbr","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7300776881539022889","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/63a420be3d8941beb8ce2380bcd31222~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lowo3hnb1apq","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205531495279099938","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/67b74808db244ad19e73380f2fd4f484~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"处的导函数"},{"type":"formula","node_id":"lowo3hnbk6g","attrs":{"formula_height":"2.791","formula_width":"9.165","image_id":"7300811606223765516","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/9f3a6a77694a4853bf8931e6adf55c6d~tplv-xv4ileqgde-image.image","value":"\\left[ f^{-1} (x)\\right]{'}"}},{"type":"text","text":"，其几何意义是曲线"},{"type":"formula","node_id":"lowo3hnbwbi","attrs":{"formula_height":"2.565","formula_width":"10.777","image_id":"7300776881539022889","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/63a420be3d8941beb8ce2380bcd31222~tplv-xv4ileqgde-image.image","value":"y=f^{-1}(x)"}},{"type":"text","text":"在点"},{"type":"formula","node_id":"lowo3hnb179w","attrs":{"formula_height":"2.262","formula_width":"6.425","image_id":"7300777920556056617","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/26508c44090549dab426dae7b66a269c~tplv-xv4ileqgde-image.image","value":"(x，y)"}},{"type":"text","text":"处的切线关于"},{"type":"formula","node_id":"lowo3hnb9ab","attrs":{"formula_height":"1.464","formula_width":"1.109","image_id":"7205531495279099938","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/67b74808db244ad19e73380f2fd4f484~tplv-xv4ileqgde-image.image","value":"y"}},{"type":"text","text":"轴正方向的斜率，如下图所示，即"},{"type":"formula","node_id":"loxrtstjztt","attrs":{"formula_height":"4.676","formula_width":"23.317","image_id":"7301131828394541097","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/01a37c2b27434ecab06e598682ca4b7e~tplv-xv4ileqgde-image.image","value":"\\frac{dy}{dx}=\\left[ f^{-1}(x)\\right]{'}=tan\\beta"}},{"type":"text","text":"。由下图显然"},{"type":"formula","node_id":"lowo3hnb1m83","attrs":{"formula_height":"4.057","formula_width":"22.427","image_id":"7300783156657913895","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/041632fb3c7842cdbb6c97dd8b3a25d9~tplv-xv4ileqgde-image.image","value":"(\\pi-\\alpha)+(\\pi-\\beta)=\\frac{\\pi}{2}"}},{"type":"text","text":"，即"},{"type":"formula","node_id":"lowo3hnbauf","attrs":{"formula_height":"4.586","formula_width":"11.928","image_id":"7300783510128803881","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/ee829d4e2ded4db5b07a405f05fc75dc~tplv-xv4ileqgde-image.image","value":"\\alpha+\\beta=\\frac{3\\pi}{2}"}},{"type":"text","text":"，那么"},{"type":"formula","node_id":"lq9n1ar1siy","attrs":{"formula_height":"5.428","formula_width":"38.567","image_id":"7313533064154759179","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8bec70ee9c0b428e83c223c67cc55780~tplv-xv4ileqgde-image.image","value":"tan\\beta=tan\\left( \\frac{3\\pi}{2}-\\alpha \\right)=cot\\alpha=\\frac{1}{tan\\alpha}"}},{"type":"text","text":"，即"},{"type":"formula","node_id":"lowo3hnbbyw","attrs":{"formula_height":"5.208","formula_width":"18.224","image_id":"7300814943841107987","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/8783f72282954a2fbafa72a2fa117657~tplv-xv4ileqgde-image.image","value":"\\left[ f^{-1} (x)\\right]'=\\frac{1}{f'(x)}"}},{"type":"text","text":"，也即"},{"type":"text","text":"反函数"},{"type":"text","text":"的"},{"type":"internal_link","node_id":"lra9xov1gcb","attrs":{"docId":"385229163457989516","target":1,"title":"导数","doc_id":"385229163457989516"},"children":[{"type":"text","text":"导数"}]},{"type":"text","text":"等于直接函数"},{"type":"text","text":"导数"},{"type":"text","text":"的"},{"type":"internal_link","node_id":"lqj1re4y1c4z","attrs":{"docId":"1935161021436582380","target":1,"title":"倒数","doc_id":"1935161021436582380"},"children":[{"type":"text","text":"倒数"}]},{"type":"reference","node_id":"lowdo81i1v5g","attrs":{"content":"lowdo81i1v5g","data_index":44,"index":44}},{"type":"reference","node_id":"loxrclitho5","attrs":{"content":"loxrclitho5","data_index":45,"index":45}},{"type":"text","text":"。"}]},{"type":"block_image","node_id":"m2349898uci","attrs":{"url":"//p9-sdbk2-sign.byteimg.com/tos-cn-i-ry4let6mk6/deebef9523f641648e3cb0e4c200586b~tplv-ry4let6mk6-image.image?lk3s=56f7592b&x-expires=1728640122&x-signature=A5vL5eDR4i0IZpOlylFpjJFQocg%3D","alt":"","note":[{"type":"paragraph","node_id":"loy31gol23e1","attrs":{},"children":[{"type":"text","text":"反函数几何意义概述图"}]}],"align":"center","width":1170,"height":936,"id":"7424060139440898060","version":"1","license":"","name":"image.png","copyright":"no_copyright","layout_width":300,"layout_height":240,"quote_user_id":"","should_transform":false}},{"type":"bullet_list","node_id":"lp2d8b7i21fw","attrs":{},"children":[{"type":"list_item","children":[{"type":"paragraph","node_id":"lp2q76tn1tar","attrs":{},"children":[{"type":"text","text":"实例"}]}]}]},{"type":"paragraph","node_id":"lp2f3ukywih","attrs":{},"children":[{"type":"text","text":"已知函数"},{"type":"formula","node_id":"lpf6ysyw22o3","attrs":{"formula_height":"2.262","formula_width":"24.927","image_id":"7305590183754203175","url":"//p3-sdbk2-media.byteimg.com/tos-cn-i-xv4ileqgde/1b7f57b95bf140289a795615cd6b905b~tplv-xv4ileqgde-image.image","value":"y=arcsinx(-1