{"id":16,"date":"2023-09-17T11:09:46","date_gmt":"2023-09-17T11:09:46","guid":{"rendered":"https:\/\/mathority.org\/id\/fungsi-invers-atau-timbal-balik\/"},"modified":"2023-09-17T11:09:46","modified_gmt":"2023-09-17T11:09:46","slug":"fungsi-invers-atau-timbal-balik","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/fungsi-invers-atau-timbal-balik\/","title":{"rendered":"Fungsi terbalik (atau timbal balik)."},"content":{"rendered":"<p>Pada artikel ini kami akan menjelaskan apa itu fungsi invers (atau timbal balik) dan cara menghitung invers suatu fungsi. Anda juga akan menemukan cara mudah mengetahui apakah suatu fungsi memiliki invers atau tidak dan sifat-sifat dari jenis fungsi tersebut. Terakhir, Anda dapat berlatih dengan latihan langkah demi langkah tentang fungsi invers. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-funcion-inversa\"><\/span> Apa fungsi inversnya?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Fungsi invers, disebut juga fungsi timbal balik, adalah fungsi yang domainnya merupakan rentang fungsi lain (fungsi asal) dan rentangnya merupakan domain fungsi asal. Fungsi invers dari fungsi <em>f<\/em> dinyatakan dengan simbol <em>f <sup>-1<\/sup><\/em> .<\/strong><\/p>\n<p> Oleh karena itu, fungsi invers dari <em>f(x)<\/em> adalah fungsi yang memenuhi kondisi berikut: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/resultat-multiplication-de-polynomes.jpg\" alt=\"fungsi invers atau timbal balik\" class=\"wp-image-685\" width=\"208\" height=\"210\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Emas<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-58f82ae2e066a1c3d7f8abc65f1a5caf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -4px;\"><\/p>\n<p> adalah fungsi kebalikan dari<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-84385e96f3e4a161ac693910846e9e0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f.\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Konsep fungsi invers juga dapat didefinisikan dengan menggunakan komposisi fungsi, karena setiap fungsi yang tersusun dengan fungsi inversnya sama dengan fungsi identitas:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8209c8ece304984d292d18d3f5968fb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f\\circ f^{-1})(x)=(f^{-1}\\circ f)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"244\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Lihat:<\/strong> <span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/id\/komposisi-fungsi-fungsi-komposit\/\">apa itu komposisi fungsi?<\/a><\/span><\/p>\n<p> Jadi kalau persamaan sebelumnya terpenuhi berarti demikian<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-58f82ae2e066a1c3d7f8abc65f1a5caf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -4px;\"><\/p>\n<p> adalah fungsi invers (atau fungsi timbal balik) dari <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-84385e96f3e4a161ac693910846e9e0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f.\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-funcion-inversa\"><\/span> Contoh fungsi invers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Mengingat definisi fungsi invers, mari kita selesaikan sebuah contoh untuk lebih memahami maknanya.<\/p>\n<ul>\n<li> Tentukan apakah fungsi-fungsi berikut merupakan invers satu sama lain:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7dbebbad962bbec034d4203171f49422_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=2x+1\\qquad g(x)=\\cfrac{x-1}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"243\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Jika kedua fungsi tersebut saling invers, maka terpenuhi 2 syarat berikut:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-facf310c96df4f7102c40db7ba7bbc5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x) = x \\qquad \\qquad  (g \\circ f)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Jadi mari kita periksa apakah kedua persamaan terpenuhi. Kita periksa dulu<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d73ab8f95080e2c6ec60813b7c9edb66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x) = x:\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36f7d5f4d815a58f808114ae471c31dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\displaystyle\\left(f \\circ g\\right)(x)&amp; = f\\Big(g(x)\\Big)\\\\[2ex]&amp;= f\\left( \\frac{x-1}{2} \\right)\\\\[2ex]&amp; = 2\\left( \\frac{x-1}{2} \\right)+1\\\\[2ex]&amp; =x-1+1\\\\[2ex]&amp;=\\bm{x} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"240\" width=\"212\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> \u27a4 Jika Anda belum memahami perhitungan yang baru saja kami lakukan, Anda perlu membuka tautan di atas untuk <em>mengetahui komposisi fungsinya?<\/em> , kami menjelaskan cara menyelesaikan jenis operasi ini dengan fungsi.<\/p>\n<p> Sehingga<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76ab7a77ded5af1f65f660a89117dd63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> ya, itu tercapai. \u2705<\/p>\n<p> Sekarang mari kita periksa kesetaraannya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42d46171e7088d03e62c95620856d280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x) = x :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-83423fe7deab472f1a7e9575ec7af8c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\left(g \\circ f\\right)(x)&amp;= g\\Big(f(x)\\Big)\\\\[2ex]&amp;= g\\Big(2x +1 \\Big)\\\\[2ex]&amp;=\\cfrac{(2x+1)-1}{2}\\\\[2ex]&amp;= \\cfrac{2x}{2}\\\\[2ex]&amp;=\\bm{x} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"249\" width=\"196\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dan kondisi invertibilitas<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f9fbc925d54530fcfa7ee7678e65b952_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> itu juga tercapai. \u2705<\/p>\n<p> Kesimpulannya, karena kedua persamaan tersebut berlaku, <strong>kedua fungsi tersebut merupakan invers satu sama lain.<\/strong><\/p>\n<p> Di bawah ini Anda dapat melihat kedua fungsi tersebut dalam grafik. Perhatikan bahwa <strong>grafik dua fungsi invers adalah simetris<\/strong> terhadap garis bagi kuadran pertama dan ketiga: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/fonction-inverse.webp\" alt=\"fungsi terbalik\" class=\"wp-image-673\" width=\"486\" height=\"402\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-saber-si-una-funcion-tiene-inversa\"><\/span> Bagaimana cara mengetahui apakah suatu fungsi mempunyai invers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Suatu fungsi memiliki fungsi invers jika merupakan fungsi injektif<\/strong> , yaitu jika setiap nilai di seluruh domainnya berkorespondensi dengan satu nilai dalam intervalnya. <\/p>\n<div class=\"wp-block-columns are-vertically-aligned-center is-layout-flex wp-container-135\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"> <span style=\"text-decoration: underline;\">Fungsi eksponensial dengan fungsi invers<\/span> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-etape-par-etape-des-fonctions-exponentielles.webp\" alt=\"\" class=\"wp-image-251\" width=\"254\" height=\"250\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow\">\n<p class=\"has-text-align-center\"> <span style=\"text-decoration: underline;\">Fungsi kuadrat tanpa fungsi invers<\/span> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/representation-d-une-fonction-quadratique-ou-parabole.webp\" alt=\"\" class=\"wp-image-137\" width=\"222\" height=\"250\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<p> Misalnya, fungsi eksponensial kiri memiliki fungsi invers karena setiap <em>x<\/em> berhubungan dengan satu nilai <em>f(x)<\/em> . Sebaliknya, fungsi kuadrat siku-siku tidak memiliki fungsi invers karena memiliki beberapa nilai <em>x<\/em> yang bayangannya sama (misalnya <em>f(1)=f(3)=2)<\/em> .<\/p>\n<p> Demikian pula fungsi bijektif terdiri dari fungsi injektif dan dugaan, oleh karena itu setiap fungsi bijektif juga mempunyai fungsi invers.<\/p>\n<p> Di sisi lain, perlu diingat bahwa invers fungsi tidak sama dengan <em>invers perkalian suatu fungsi<\/em> , melainkan dua konsep yang berbeda. Untuk mencari invers perkalian suatu fungsi, cukup hitung 1 korespondensi dengan fungsi tersebut.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4bf5aff2d6025747e58082ac697b06d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f^{-1}(x) \\neq \\cfrac{1}{f(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"113\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Pada bagian selanjutnya kita akan melihat cara menentukan fungsi invers. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hallar-la-funcion-inversa\"><\/span> Cara mencari fungsi invers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Untuk menghitung invers suatu fungsi, langkah-langkah berikut harus dilakukan: <\/p>\n<div style=\"padding-top: 30px; padding-bottom: 10px; padding-right: 30px; padding-left: 15px; border: 2px dashed #FF9B28; border-radius:30px; margin-bottom:25px\">\n<ol style=\"color:#FF8A05; font-weight: bold;border:\">\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Gantikan <em>f(x)<\/em> dengan <em>y<\/em> .<\/span><\/li>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Ubah semua <em>x<\/em> menjadi <em>y<\/em> , dan sebaliknya.<\/span><\/li>\n<li style=\"margin-bottom:12px\"> <span style=\"color:#101010;font-weight: normal;\">Hapus variabel <em>y<\/em> .<\/span><\/li>\n<li> <span style=\"color:#101010;font-weight: normal;\">Gantikan variabel <em>y<\/em> dengan <em>f <sup>-1<\/sup> (x)<\/em> . Fungsi invers adalah ekspresi yang ditemukan untuk <em>f <sup>-1<\/sup> (x)<\/em> .<\/span><\/li>\n<\/ol>\n<\/div>\n<p> Agar Anda dapat melihat secara pasti bagaimana cara menghitung invers fungsi, kita akan menentukan invers fungsi berikut sebagai contoh:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7bd000cffbbb2432fd2c53b683d15c5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x) =4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Pertama-tama, kita perlu menggantinya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Untuk<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-824b16de72dde879834460d93bd88610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=4x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Sekarang kami mengubah segalanya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> dari fungsi oleh<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , dan sebaliknya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a9c604955a9823146b45d8d7fcf955d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y= 4x+5 \\quad \\xrightarrow{x \\ \\rightarrow \\ y \\ ; \\ y \\ \\rightarrow \\ x} \\quad x= 4y+5\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"317\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Kemudian kita menghapus variabelnya <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a67252387800a0b4cf9d5de3f47992ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=4y+5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c5c3b6dd8da6675c67d8d97143338bc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x-5=4y\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d3b7fce291e0a2e8783d43d7bf3091dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x-5}{4}=y\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"73\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-94afb1cafad9006eb47ba582cf925cb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{x-5}{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Dan terakhir, fungsi kebalikan dari<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> adalah ekspresi aljabar yang kita peroleh dengan mengisolasi <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c678d462718983cd6440a900898d488b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{f^{-1}(x) = } \\cfrac{\\bm{x-5}}{\\bm{4}}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"115\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-funcion-inversa\"><\/span> Soal latihan fungsi invers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Di bawah ini kami telah menyiapkan beberapa latihan langkah demi langkah tentang fungsi invers agar Anda dapat berlatih.<\/p>\n<p> \ud83d\udc49 Ingatlah bahwa jika Anda tidak memahami cara menyelesaikan suatu latihan atau ingin kami menyelesaikan masalah untuk Anda, Anda dapat menulis kepada kami di komentar!<\/p>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Periksa apakah dua fungsi berikut invers (atau timbal balik) atau tidak: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b15be151088ebf4c65087b5f4e00938_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=3x-7\\qquad g(x)=\\cfrac{x+7}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"243\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Agar kedua fungsi tersebut invers satu sama lain, hal berikut ini harus benar:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f64196e1a5f56f51ec1b1164eceb60b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x)=x \\qquad \\qquad (g \\circ f)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu perlu dilakukan pengecekan apakah kedua syarat tersebut terpenuhi. Kita periksa dulu <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-11b7c4f21f3fb296e8a27a20cf8d1c3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x)=x :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fbbdf5bb2c308a1973c5f2d99e822ad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\displaystyle\\left(f \\circ g\\right)(x)&amp;= f\\Big(g(x)\\Big)\\\\[2ex]&amp;= f\\left( \\frac{x+7}{3} \\right)\\\\[2ex]&amp;= 3 \\left(\\frac{x +7}{3} \\right) - 7  \\\\[2ex] &amp; =x + 7 - 7 \\\\[2ex]&amp;= \\bm{x}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"240\" width=\"213\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Belum,<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76ab7a77ded5af1f65f660a89117dd63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> ya, itu tercapai. \u2705<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita periksa komposisi fungsi lainnya <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7b058926aba155b6b765d5724492a92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x)=x :\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac5819584c8358b812c8214a902e4f08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\left(g \\circ f\\right)(x)&amp;= g\\Big(f(x)\\Big)\\\\[2ex]&amp;= g\\left(3x-7\\right)\\\\[2ex]&amp;=\\cfrac{(3x-7)+7}{3}\\\\[2ex]&amp;=\\cfrac{3x}{3}\\\\[2ex]&amp;=\\bm{x}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"239\" width=\"196\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dimana<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f9fbc925d54530fcfa7ee7678e65b952_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x) = x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> itu juga tercapai. \u2705<\/p>\n<p class=\"has-text-align-left\"> Bagaimana hal itu bisa terjadi<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d3c53166029da197c6e17e19d3809563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f \\circ g)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> Dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-db26718e178acd21a2f1f266b9a3c3b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(g \\circ f)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<p> , kedua fungsi tersebut berbanding terbalik satu sama lain.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 2<\/h3>\n<p> Hitung invers (atau fungsi timbal balik) dari fungsi polinomial derajat pertama berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e230d2b35a28b2bdfc77f06634de138b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=-2x-3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Hal pertama yang harus dilakukan untuk membalikkan fungsi adalah dengan mengganti sukunya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Untuk<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a4a9dab43d26aecc4bb0ce18ea508c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"}y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b754ae8eb949f8f8d85ba36ebabb52d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=-2x-3\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita mengubah<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> oleh<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , dan sebaliknya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-653d447dc0dcd501486c99f8cb1d40f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=-2x-3 \\ \\xrightarrow{x \\ \\rightarrow \\ y \\ ; \\ y \\ \\rightarrow \\ x} \\ x =-2y-3\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"322\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kami melepaskannya <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42c5d877f3a0a6fd971e65ade72afdb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x =-2y-3\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23c71b5b09a6e2cef29e449dcd1efa16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x +3=-2y\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9611a1a60c491236193f13a0542011bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x +3}{-2}=y\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"73\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b026add174bb871e220df0191afdbd76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = \\cfrac{x +3}{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami sudah berhasil merilisnya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> . Oleh karena itu, kebalikan dari fungsi<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Timur: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2197f512ad2a7360c966f64b78e75d53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{f^{-1}(x) = -} \\cfrac{\\bm{x+3}}{\\bm{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"133\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Latihan 3<\/h3>\n<p> Balikkan fungsi polinomial kuadrat berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8200d353eb8e21c4c04f3601fc802489_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)=x^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Untuk mencari fungsi invers kita akan mengikuti prosedur yang kita lihat di atas. Jadi kami akan menelepon<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> ke fungsinya<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40e51e628d64bea41578e16139b71b6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d47514334dca13875d8344a469e1c7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=x^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kedua, kami memodifikasi<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Untuk<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , dan sebaliknya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-29de23ff2053fc75656c8ff4d7fa41a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=x^2-1 \\ \\xrightarrow{x \\ \\rightarrow \\ y \\ ; \\ y \\ \\rightarrow \\ x} \\ x =y^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"291\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan terakhir, kami mengisolasi variabelnya <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e69105ebcca11f4a74f3f3657e3e6440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x =y^2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5728627e7e38f4515904dee980b00694_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x+1=y^2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1cccfbd1b731987173cb8242913f3015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt{x+1}=y\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a8de5ae78dde4cbb3d501a56dec92e6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\pm \\sqrt{x+1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Namun dalam hal ini fungsi yang diperoleh memiliki dua citra untuk setiap elemen domainnya (citra positif dan citra negatif). Oleh karena itu, <strong>tidak ada fungsi invers<\/strong> dari fungsi masalah.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 4<\/h3>\n<p> Tentukan fungsi invers (atau fungsi timbal balik) dari fungsi rasional berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a7ccc4e4bbef9c4997233bfc9271bf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle f(x)=\\frac{x-1}{2x+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"110\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E6F9EF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E6F9EF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Lihat solusinya<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama, kita ganti<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Untuk<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-15a4196ec4380923616fae6f09302cb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{x-1}{2x+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"85\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita mengubah<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> pembilang dan penyebut<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , dan sebaliknya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bf93d67efcb45f42b278c4bbdd134482_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{x-1}{2x+3} \\ \\xrightarrow{x \\ \\rightarrow \\ y \\ ; \\ y \\ \\rightarrow \\ x} \\ x=\\cfrac{y-1}{2y+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"298\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kami melepaskannya <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e5501ae45c059ce0992ae626ae90dccd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{y-1}{2y+3}\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"85\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ekspresi<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3535e891270ab3ae4ed4aad6a498984_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2y +3\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -4px;\"><\/p>\n<p> membagi seluruh ruas kanan persamaan, sehingga kita dapat mengalikannya dengan mengalikan seluruh ruas kiri persamaan: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6a7e58aaef340adec944945b7feb0f8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\\cdot (2y+3)=y-1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b837ac50ef3677cc95e0a2d439a76d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2yx+3x=y-1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"132\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menyetujui semua persyaratan<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> di satu sisi persamaan, dan suku lain di sisi lain:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-525563a5b84daff8946bfe896e5d30d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2yx-y=-3x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"145\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk membersihkan<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42796b3d2f8b77c564fefa868de8d64a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"}y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , kita mengekstrak faktor persekutuan dari ruas kiri persamaan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d21c3a81a922f2d84a75a4568b97dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y(2x-1)=-3x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"159\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan sebagai tukang pos<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cbbee5c6fcee058c32f9f067554f4679_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(2x-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -5px;\"><\/p>\n<p> adalah mengalikan seluruh ruas kiri persamaan, kita dapat melakukannya dengan membagi seluruh ruas kanan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5ac1a61f0a3683aa8cde1c6ad21a1cae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{-3x-1}{2x-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"107\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami sudah berhasil merilisnya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> . Jadi fungsi invers dari<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7ee323bc5a3f73ad5e066b13bed5504_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"34\" style=\"vertical-align: -5px;\"><\/p>\n<p> Timur: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40fd07fe81e86ffb37c7b857f43a140f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{f^{-1}(x)=} \\cfrac{\\bm{-3x-1}}{\\bm{2x-1}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"138\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-de-la-funcion-inversa\"><\/span> Sifat-sifat fungsi invers<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Fungsi invers mempunyai ciri-ciri sebagai berikut:<\/p>\n<ul>\n<li> Fungsi invers bersifat unik, yaitu jika suatu fungsi dapat dibalik, maka untuk fungsi tersebut hanya terdapat satu fungsi invers.<\/li>\n<\/ul>\n<ul>\n<li> Domain dari fungsi invers adalah jangkauan (atau jangkauan) dari fungsi aslinya.<\/li>\n<\/ul>\n<ul>\n<li> Demikian pula, jalur fungsi invers ekuivalen dengan domain fungsi aslinya.<\/li>\n<\/ul>\n<ul>\n<li> Setiap fungsi yang tersusun dari fungsi inversnya menghasilkan fungsi identitas (x).<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8209c8ece304984d292d18d3f5968fb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f\\circ f^{-1})(x)=(f^{-1}\\circ f)(x)=x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"244\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Grafik suatu fungsi dan grafik invers fungsi tersebut simetris terhadap garis bagi kuadran pertama dan ketiga.<\/li>\n<\/ul>\n<ul>\n<li> Invers dari fungsi invers sama dengan fungsi aslinya:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-53db614ba7e0fdd5be384f53e05c6d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(f^{-1}\\right)^{-1}=f\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"95\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> Membalik fungsi komposit sama dengan menghitung invers setiap fungsi secara terpisah dan kemudian menyusun fungsi inversnya.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e5f9cc891e8322cf737caf8fa76a92f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(f\\circ g)^{-1}=f^{-1}\\circ g^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Suatu fungsi kontinu secara simultan dengan fungsi inversnya, atau dengan kata lain jika suatu fungsi kontinu maka invers fungsi tersebut juga demikian.<\/li>\n<\/ul>\n<ul>\n<li> Jika suatu fungsi terdiferensiasi dan turunannya tidak hilang sewaktu-waktu\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7e8b3cc99b3a65d07a0bfbdd9680062_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"f'(x)\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<p> , fungsi inversnya juga akan terdiferensiasi.<\/li>\n<\/ul>\n<p> Selain itu, turunan fungsi invers dapat dihitung dengan menerapkan <strong>teorema fungsi invers<\/strong> , yang rumusnya adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1aa57f06b0b96e0900dd47e0bf4ad66c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(f^{-1}\\right)'(y)=\\cfrac{1}{f'(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"139\" style=\"vertical-align: -17px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Pada artikel ini kami akan menjelaskan apa itu fungsi invers (atau timbal balik) dan cara menghitung invers suatu fungsi. Anda juga akan menemukan cara mudah mengetahui apakah suatu fungsi memiliki invers atau tidak dan sifat-sifat dari jenis fungsi tersebut. Terakhir, Anda dapat berlatih dengan latihan langkah demi langkah tentang fungsi invers. Apa fungsi inversnya? Fungsi &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/fungsi-invers-atau-timbal-balik\/\"> <span class=\"screen-reader-text\">Fungsi terbalik (atau timbal balik).<\/span> Selengkapnya &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[49],"tags":[],"class_list":["post-16","post","type-post","status-publish","format-standard","hentry","category-representasi-fungsi"],"yoast_head":"<!-- This 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