Invers Kebalikan dalam Fungsi Matematika Bersama Contohnya

Pinter Pandai

8 tahun ago

Fungsi Invers Kebalikan dalam Matematika

Fungsi Invers (atau fungsi invers kebalikan) adalah (dalam matematika) fungsi yang merupakan kebalikan aksi dari suatu fungsi.

Contohnya anggap saja $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" alt="{\displaystyle f}" width="550.5" height="1080.4">$ sebuah fungsi dari himpunan A ke himpunan B.

Bila dapat ditentukan sebuah fungsi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" alt="{\displaystyle g}" width="480.5" height="865.1">$ dari himpunan B ke himpunan A sedemikian, sehingga $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ba371f5aa81db76dceea60e30a1a0cc412da77" alt="{\displaystyle g(f(a))=a}" width="4982.1" height="1223.9">$ dan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649edce920742e1030c8c3254835c10aa80ed749" alt="{\displaystyle f(f(b))=b}" width="4852.1" height="1223.9">$ untuk setiap a dalam A dan b dalam B, maka $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" alt="{\displaystyle g}" width="480.5" height="865.1">$ disebut fungsi invers dari $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" alt="{\displaystyle f}" width="550.5" height="1080.4">$ dan bisa ditulis sebagai $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" alt="{\displaystyle f^{-1}}" width="1572.9" height="1295.7">$ .

Fungsi invers kebalikan dalam matematika.

Sebelum mengetahui fungsi ini maka harus mengenali dahulu fungsi yang memiliki invers.

Fungsi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" alt="{\displaystyle f(x)}" width="1902" height="1223.9">$ akan memiliki invers kebalikan dengan syarat $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" alt="{\displaystyle f(x)}" width="1902" height="1223.9">$ merupakan fungsi bijektif.

Jika fungsi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" alt="{\displaystyle f}" width="550.5" height="1080.4">$ memetakan anggota himpunan A ke himpunan B maka invers dari fungsi $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" alt="{\displaystyle f}" width="550.5" height="1080.4">$ atau ditulis $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" alt="{\displaystyle f^{-1}}" width="1572.9" height="1295.7">$ memetakan himpunan B ke himpunan A.

Kemudian ketika suatu bilangan itu dioperasikan dengan inversnya, maka akan menghasilkan identitas.

Identitas adalah suatu bilangan yang jika dioperasikan dengan suatu bilangan, maka akan menghasilkan suatu bilangan tersebut dan pada operasi perkalian, identitasnya adalah 1 karena apabila dikalikan dengan suatu bilangan hasilnya suatu bilangan.

Sedangkan, pada penjumlahan identitasnya adalah 0 karena bila dijumlahkan dengan bilangan tertentu hasilnya bilangan tertentu.

Pada fungsi juga berlaku demikian, sebuah fungsi bila dikomposisikan dengan invers maka menghasilkan fungsi identitas, yaitu $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f690285952308aa49e3c6aac892df31cad6d1b06" alt="{\displaystyle f(x)=x}" width="3808.6" height="1223.9">$ .

Contoh Fungsi Invers Kebalikan

Tentukan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3c6d4bd6502539cdca9cf8dd58eeedc1345aad" alt="{\displaystyle f(x)=2x+3}" width="6032.5" height="1223.9">$ !

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3c6d4bd6502539cdca9cf8dd58eeedc1345aad" alt="{\displaystyle f(x)=2x+3}" width="6032.5" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd970ca3ba35c94fd63ee32e42e62ccdec01e79f" alt="{\displaystyle f(x)-3=2x}" width="6032.5" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b95191b90fdd6a4e41f02e02ab44457bb9c29c9" alt="{\displaystyle x={\frac {f(x)-3}{2}}}" width="5892" height="2443.8">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21470f6678e7a0e7c603f5926015d1ff3b5e08da" alt="{\displaystyle f^{-1}(x)={\frac {x-3}{2}}}" width="6914.4" height="2228.5">

Tentukan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2235989716c469c63f40fd9a7909b0e90e6e986b" alt="{\displaystyle f(x)=x^{2}-6x+15}" width="8782.4" height="1367.4">$ !

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2235989716c469c63f40fd9a7909b0e90e6e986b" alt="{\displaystyle f(x)=x^{2}-6x+15}" width="8782.4" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a130afcad8134c1e71669c722eb8268095229b" alt="{\displaystyle f(x)=x^{2}-6x+9-9+15}" width="12229.2" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9551529e424da6f5e012447afa2482b99359249d" alt="{\displaystyle f(x)=(x+3)^{2}+6}" width="8488.4" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bee4e93d0869699ade4e626877aee9d173a50f2" alt="{\displaystyle f(x)-6=(x+3)^{2}}" width="8488.4" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/554e9eb37c85536ea7129baf4df2f78d700a420b" alt="{\displaystyle x+3=\pm {\sqrt {f(x)-6}}}" width="9034.4" height="2085">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00415bf235b2195a7748829194917506f434dc1a" alt="{\displaystyle x=-3\pm {\sqrt {f(x)-6}}}" width="9034.4" height="2085">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/855805791c7c59b539ab335e75ec6806462b40d7" alt="{\displaystyle f^{-1}(x)=-3\pm {\sqrt {x-6}}}" width="9889.9" height="1367.4">

Tentukan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2775cdb69b81f32fb1513a34420678e5bd8969da" alt="{\displaystyle f(x)={\frac {2x-7}{5x+3}}}" width="6392.5" height="2300.3">$ !

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2775cdb69b81f32fb1513a34420678e5bd8969da" alt="{\displaystyle f(x)={\frac {2x-7}{5x+3}}}" width="6392.5" height="2300.3">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae9a7032bb3b1bb71e37ba24b1ccbc0bd629a682" alt="{\displaystyle (5x+3)f(x)=2x-7}" width="9607.9" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c692705ecf4c455a2568bdeddafdeb1df3a7d0d" alt="{\displaystyle 5xf(x)+3f(x)=2x-7}" width="10730.9" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c557df5134f958c7debcf7c18c73edf29ff3750" alt="{\displaystyle 5xf(x)-2x=-3f(x)-7}" width="11509.4" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558522c4d27b55b30a78637122c530d03110a7d3" alt="{\displaystyle (5f(x)-2)x=-3f(x)-7}" width="11715.9" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d0873c8457d6112227acf9982165a8c0aee9c8" alt="{\displaystyle x={\frac {-3f(x)-7}{5f(x)-2}}}" width="7171" height="2802.6">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66ba57a72945a8972e3665d297f0061b8decb918" alt="{\displaystyle f^{-1}(x)={\frac {-3x-7}{5x-2}}}" width="8193.4" height="2300.3">

Tentukan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf883ebbebbfa47e4917acb9eba93cf5466074c5" alt="{\displaystyle f(x)=e^{x+3}}" width="5111.8" height="1367.4">$ !

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf883ebbebbfa47e4917acb9eba93cf5466074c5" alt="{\displaystyle f(x)=e^{x+3}}" width="5111.8" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bbba6ad495b8ccc7257dc0dd5cea37e3bef9e09" alt="{\displaystyle ^{e}\log f(x)=x+3}" width="7574.7" height="1223.9">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e6e3ea0c879a02c844d9fafa9da16a132988be" alt="{\displaystyle x=^{e}\log f(x)-3}" width="7408" height="1223.9">

(karena

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d1d79399f48049ec69aa72999b0d1200da4465f" alt="{\displaystyle ^{e}\log x=\ln x}" width="5523.4" height="1152.1">

)

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d80536ef0481c14773ac453a64af1c762115d790" alt="{\displaystyle f^{-1}(x)=\ln x-3}" width="7556.1" height="1367.4">

Tentukan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6fb133ec2407f22ec0078797bd7b4f7ca7f452" alt="{\displaystyle f(x)=^{2}\log {(x^{2}+8x-25)}}" width="11461.4" height="1367.4">$ !

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6fb133ec2407f22ec0078797bd7b4f7ca7f452" alt="{\displaystyle f(x)=^{2}\log {(x^{2}+8x-25)}}" width="11461.4" height="1367.4">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddad2f04a0171cbb8de4610750fc87262a123de9" alt="{\displaystyle x^{2}+8x+16-16-25=2^{f(x)}}" width="13273.7" height="1295.7">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb8d4ef6ebfcd4f5a61cd390a0a750721d4c9e5" alt="{\displaystyle x^{2}+8x+16-41=2^{f(x)}}" width="11049.7" height="1295.7">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/127191fc7503b832efe0380460b078dfb1e71762" alt="{\displaystyle x+4=\pm {\sqrt {41+2^{f(x)}}}}" width="9578.4" height="2085">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49471e4d7ecbc1a7f7faacf18e0b99492c55a0f0" alt="{\displaystyle x=-4\pm {\sqrt {41+2^{f(x)}}}}" width="9578.4" height="2085">

<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/548a20559654780ee792e60c5f3552aea848f67c" alt="{\displaystyle f^{-1}(x)=-4\pm {\sqrt {41+2^{x}}}}" width="10990.2" height="1510.9">

Tentukan $<img decoding="async" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c1727d29d2376fdab03f89a8ae0f8c7d2a0856" alt="{\displaystyle f(x)=sin(4x+3)-7}" width="9950.4" height="1223.9">$ !