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Diketahui , . Jika adalah invers fungsi , maka

Pertanyaan

Diketahui begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 2 x plus 1 over denominator x minus 3 end fraction end style, begin mathsize 14px style x not equal to 3 end style. Jika begin mathsize 14px style f to the power of negative 1 end exponent end style adalah invers fungsi begin mathsize 14px style f end style, maka begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x minus 2 close parentheses equals.... end style 

Pembahasan Soal:

Untuk mencari invers dari fungsi begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 2 x plus 1 over denominator x minus 3 end fraction end style, pertama kita ubah bentuk begin mathsize 14px style y equals f open parentheses x close parentheses end style menjadi begin mathsize 14px style x equals f open parentheses y close parentheses end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator 2 x plus 1 over denominator x minus 3 end fraction end cell row cell y open parentheses x minus 3 close parentheses end cell equals cell 2 x plus 1 end cell row cell x y minus 3 y end cell equals cell 2 x plus 1 end cell row cell x y minus 2 x end cell equals cell 1 plus 3 y end cell row cell x open parentheses y minus 2 close parentheses end cell equals cell 1 plus 3 y end cell row x equals cell fraction numerator 1 plus 3 y over denominator y minus 2 end fraction end cell end table end style

Lalu ganti begin mathsize 14px style x end style sebagai begin mathsize 14px style f to the power of negative 1 end exponent open parentheses y close parentheses end style sehingga begin mathsize 14px style f to the power of negative 1 end exponent open parentheses y close parentheses equals f open parentheses y close parentheses end style.

begin mathsize 14px style f to the power of negative 1 end exponent open parentheses y close parentheses equals fraction numerator 1 plus 3 y over denominator y minus 2 end fraction end style

Kemudian ubah variabel begin mathsize 14px style y end style dengan begin mathsize 14px style x minus 2 end style sehingga didapat begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x minus 2 close parentheses end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f to the power of negative 1 end exponent open parentheses x minus 2 close parentheses end cell equals cell fraction numerator 1 plus 3 open parentheses x minus 2 close parentheses over denominator open parentheses x minus 2 close parentheses minus 2 end fraction end cell row blank equals cell fraction numerator 1 plus 3 x minus 6 over denominator x minus 2 minus 2 end fraction end cell row blank equals cell fraction numerator 3 x minus 5 over denominator x minus 4 end fraction end cell end table end style

Jadi, begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x minus 2 close parentheses equals fraction numerator 3 x minus 5 over denominator x minus 4 end fraction end style.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

F. Kartikasari

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 29 Maret 2021

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Pertanyaan yang serupa

Diketahui fungsi Invers fungsi f(x) adalah

Pembahasan Soal:

f open parentheses x close parentheses equals fraction numerator 5 x plus 2 over denominator 3 x minus 1 end fraction  3 f open parentheses x close parentheses x minus f open parentheses x close parentheses equals 5 x plus 2  3 f open parentheses x close parentheses x minus 5 x equals f open parentheses x close parentheses plus 2  x left parenthesis 3 f open parentheses x close parentheses minus 5 right parenthesis equals f open parentheses x close parentheses plus 2  space x equals fraction numerator f open parentheses x close parentheses plus 2 over denominator 3 f open parentheses x close parentheses minus 5 end fraction  f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x plus 2 over denominator 3 x minus 5 end fraction

0

Roboguru

Fungsi  dan  ditentukan dengan  dan , maka  = ...

Pembahasan Soal:

Menentukan g left parenthesis x right parenthesis

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell equals cell fraction numerator x minus 3 over denominator 2 x end fraction end cell row cell g left parenthesis x right parenthesis to the power of negative 1 end exponent end cell equals cell fraction numerator x minus 3 over denominator 2 x end fraction end cell row cell fraction numerator 1 over denominator g left parenthesis x right parenthesis end fraction end cell equals cell fraction numerator x minus 3 over denominator 2 x end fraction end cell row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x over denominator x minus 3 end fraction semicolon space x not equal to 3 end cell end table 

Menentukan g to the power of negative 1 end exponent left parenthesis x right parenthesis 

table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x over denominator x minus 3 end fraction end cell row y equals cell fraction numerator 2 x over denominator x minus 3 end fraction end cell row cell y left parenthesis x minus 3 right parenthesis end cell equals cell 2 x end cell row cell x y minus 3 y end cell equals cell 2 x end cell row cell x y minus 2 x end cell equals cell 3 y end cell row cell x left parenthesis y minus 2 right parenthesis end cell equals y row x equals cell fraction numerator y over denominator y minus 2 end fraction end cell row cell g to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell fraction numerator y over denominator y minus 2 end fraction end cell row cell g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator x over denominator x minus 2 end fraction comma space x not equal to 2 end cell end table 

Dengan demikian, diperoleh g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x over denominator x minus 2 end fraction comma space x not equal to 2.

0

Roboguru

Diketahui jika fungsi  maka invers dari fungsi  adalah ...

Pembahasan Soal:

Ingat definisi dari fungsi invers atau fungsi kebalikan yaitu suatu fungsi yang berkebalikan dari fungsi asalnya. Suatu fungsi f memiliki fungsi invers f to the power of negative 1 end exponent jika blank f open parentheses x close parentheses merupakan fungsi satu-satu dan fungsi pada (bijektif), atau biasa ditulis y equals blank f open parentheses x close parentheses left right arrow x equals f to the power of negative 1 end exponent left parenthesis y right parenthesis. Sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell fraction numerator 5 minus 3 x over denominator 6 x minus 2 end fraction end cell row y equals cell fraction numerator 5 minus 3 x over denominator 6 x minus 2 end fraction end cell row cell y left parenthesis 6 x minus 2 right parenthesis end cell equals cell 5 minus 3 x end cell row cell 6 x y minus 2 y end cell equals cell 5 minus 3 x end cell row cell 6 x y plus 3 x end cell equals cell 5 plus 2 y end cell row cell x left parenthesis 6 y plus 3 right parenthesis end cell equals cell 5 plus 2 y end cell row x equals cell fraction numerator 5 plus 2 y over denominator 6 y plus 3 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator 5 plus 2 y over denominator 6 y plus 3 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 5 plus 2 x over denominator 6 x plus 3 end fraction end cell end table

Jadi, dapat disimpulkan bahwa invers dari f open parentheses x close parentheses equals fraction numerator 5 minus 3 x over denominator 6 x minus 2 end fraction adalah f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 5 plus 2 x over denominator 6 x plus 3 end fraction.

0

Roboguru

Tentukan rumus fungsi invers tiap fungsi berikut. a.

Pembahasan Soal:

begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 2 over denominator 2 x minus 4 end fraction comma space x not equal to 2 end style 

Misalkan begin mathsize 14px style f open parentheses x close parentheses equals y end style, maka:

table attributes columnalign right center left columnspacing 2px end attributes row cell fraction numerator size 14px 2 over denominator size 14px 2 size 14px x size 14px minus size 14px 4 end fraction end cell size 14px equals size 14px y row size 14px 2 size 14px equals cell size 14px y begin mathsize 14px style left parenthesis 2 x minus 4 right parenthesis end style end cell row cell size 14px 2 over size 14px y end cell size 14px equals cell size 14px 2 size 14px x size 14px minus size 14px 4 end cell row cell size 14px 2 over size 14px y size 14px plus size 14px 4 end cell size 14px equals cell size 14px 2 size 14px x end cell row x size 14px equals cell size 14px 1 over size 14px y size 14px plus size 14px 2 size 14px space end cell row cell size 14px f to the power of size 14px minus size 14px 1 end exponent begin mathsize 14px style left parenthesis y right parenthesis end style end cell size 14px equals cell size 14px 1 over size 14px y size 14px plus size 14px 2 size 14px semicolon size 14px space size 14px y size 14px not equal to size 14px 0 end cell end table

Jadi, rumus fungsi invers begin mathsize 14px style f end style adalah begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses equals 1 over x plus 2 space comma space x not equal to 0 end style.

0

Roboguru

Jika , maka  adalah ...

Pembahasan Soal:

Diketahui:

f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator 3 x plus 4 end fraction comma space x not equal to negative 4 over 3

Ditanya:

f to the power of negative 1 end exponent left parenthesis x right parenthesis

Terdapat 3 langkah untuk menentukan fungsi invers, yaitu:

  1. Ubahlah bentuk y equals f open parentheses x close parentheses menjadi bentuk x equals f open parentheses y close parentheses.
  2. Tuliskan x sebagai f to the power of negative 1 end exponent open parentheses y close parentheses sehingga f to the power of negative 1 end exponent open parentheses y close parentheses equals f open parentheses y close parentheses.
  3. Ubahlah variabel y dengan x sehingga diperoleh rumus fungsi invers f to the power of negative 1 end exponent left parenthesis x right parenthesis.

Invers fungsi f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator 3 x plus 4 end fraction:

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell fraction numerator 2 x minus 1 over denominator 3 x plus 4 end fraction end cell row y equals cell fraction numerator 2 x minus 1 over denominator 3 x plus 4 end fraction end cell row cell y open parentheses 3 x plus 4 close parentheses end cell equals cell open parentheses 2 x minus 1 close parentheses end cell row cell 3 x y plus 4 y end cell equals cell 2 x minus 1 end cell row cell 3 x y minus 2 x end cell equals cell negative 4 y minus 1 end cell row cell x open parentheses 3 y minus 2 close parentheses end cell equals cell negative 4 y minus 1 end cell row x equals cell fraction numerator negative 4 y minus 1 over denominator 3 y minus 2 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator negative 4 y minus 1 over denominator 3 y minus 2 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 4 x minus 1 over denominator 3 x minus 2 end fraction end cell end table 

Jadi, fungsi invers dari fungsi f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator 3 x plus 4 end fraction comma space x not equal to negative 4 over 3 adalah Error converting from MathML to accessible text..

0

Roboguru

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