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Diketahui , dan . Tentukan: c. ,

Pertanyaan

Diketahui f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 1 end fraction comma space g open parentheses x close parentheses equals x squared minus 1, dan h open parentheses x close parentheses equals 3 x plus 5. Tentukan:

c. open parentheses h to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses

Pembahasan Soal:

Ingat kembali:

open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses 

Berarti:

open parentheses h to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals open parentheses f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses  

Diketahui f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 1 end fraction comma space g open parentheses x close parentheses equals x squared minus 1, dan h open parentheses x close parentheses equals 3 x plus 5. Maka:

open parentheses f ring operator h close parentheses open parentheses x close parentheses equals f open parentheses h open parentheses x close parentheses close parentheses open parentheses f ring operator h close parentheses open parentheses x close parentheses equals f open parentheses 3 x plus 5 close parentheses open parentheses f ring operator h close parentheses open parentheses x close parentheses equals fraction numerator 2 open parentheses 3 x plus 5 close parentheses minus 1 over denominator open parentheses 3 x plus 5 close parentheses plus 1 end fraction open parentheses f ring operator h close parentheses open parentheses x close parentheses equals fraction numerator 6 x plus 10 minus 1 over denominator 3 x plus 5 plus 1 end fraction open parentheses f ring operator h close parentheses open parentheses x close parentheses equals fraction numerator 6 x plus 9 over denominator 3 x plus 6 end fraction 

Sehingga:

 space space space space open parentheses f ring operator h close parentheses open parentheses x close parentheses equals fraction numerator 6 x plus 9 over denominator 3 x plus 6 end fraction space space space space space space space space space space space space space space space y equals fraction numerator 6 x plus 9 over denominator 3 x plus 6 end fraction space y times open parentheses 3 x plus 6 close parentheses equals 6 x plus 9 space space space space 3 x y plus 6 y equals 6 x plus 9 space space space space 3 x y minus 6 x equals 9 minus 6 y space space space x open parentheses 3 y minus 6 close parentheses equals 9 minus 6 y space space space space space space space space space space space space space space space x equals fraction numerator 9 minus 6 y over denominator 3 y minus 6 end fraction open parentheses f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 9 minus 6 x over denominator 3 x minus 6 end fraction 

Jadi, open parentheses h to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator 9 minus 6 x over denominator 3 x minus 6 end fraction.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

W. Lestari

Mahasiswa/Alumni Universitas Sriwijaya

Terakhir diupdate 02 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Fungsi  dan  ditentukan oleh  dan . Rumus untuk  adalah ...

Pembahasan Soal:

Diketahui:

Fungsi begin mathsize 14px style f open parentheses x close parentheses equals 3 x minus 2 end style dan begin mathsize 14px style g open parentheses x close parentheses equals x plus 5 end style.

Tentukan terlebih dahulu fungsi komposisi begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses end style.

Fungsi komposisi merupakan suatu penggabungan dari suatu operasi dua jenis fungsi begin mathsize 14px style f open parentheses x close parentheses end style dan juga begin mathsize 14px style g open parentheses x close parentheses end style sehingga mampu menghasilkan suatu fungsi baru.

Diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell g open parentheses 3 x minus 2 close parentheses end cell row blank equals cell open parentheses 3 x minus 2 close parentheses plus 5 end cell row blank equals cell 3 x plus 3 end cell end table end style

Kemudian tentukan fungsi invers komposisi begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses end style atau begin mathsize 14px style open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style.

Fungsi invers atau fungsi kebalikan merupakan suatu fungsi yang berkebalikan dari fungsi asalnya. 

Diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell 3 x plus 3 end cell row y equals cell 3 x plus 3 end cell row cell 3 x end cell equals cell y minus 3 end cell row x equals cell fraction numerator y minus 3 over denominator 3 end fraction end cell row x equals cell 1 third y minus 1 end cell row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell 1 third y minus 1 end cell row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell 1 third x minus 1 end cell end table end style

Rumus untuk begin mathsize 14px style open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style adalah begin mathsize 14px style 1 third x minus 1 end style.

Roboguru

Diketahui fungsi  dan  dirumuskan oleh  dan . Jika . Tentukanlah invers .

Pembahasan Soal:

Diketahui fungsi f left parenthesis x right parenthesis equals fraction numerator 2 x plus 1 over denominator x minus 2 end fraction comma space x not equal to 2 dan g left parenthesis x right parenthesis equals x plus 3. Invers dari masing-masing fungsi tersebut adalah

f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative d x plus b over denominator c x minus a end fraction equals fraction numerator negative left parenthesis negative 2 right parenthesis x plus 1 over denominator x minus 2 end fraction equals fraction numerator 2 x plus 1 over denominator x minus 2 end fraction, dan

g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x minus b over denominator a end fraction equals fraction numerator x minus 3 over denominator 1 end fraction equals x minus 3.

Jika h left parenthesis x right parenthesis equals open parentheses f ring operator g close parentheses left parenthesis x right parenthesis, maka

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

Jadi, invers dari fungsi h left parenthesis x right parenthesis adalah h to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative x plus 3 over denominator x minus 2 end fraction.

Roboguru

Jika diketahui fungsi  = ...

Pembahasan Soal:

Penyelesaian:

f left parenthesis x right parenthesis equals fraction numerator 2 x minus 3 over denominator 4 x plus 5 end fraction  f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative 5 x minus 3 over denominator 4 x minus 2 end fraction  g left parenthesis x right parenthesis equals square root of x minus 1 end root  y equals square root of x minus 1 end root  y squared equals x minus 1  x equals y squared plus 1  g to the power of negative 1 end exponent left parenthesis x right parenthesis equals x squared plus 1    left parenthesis f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis equals f to the power of negative 1 end exponent left parenthesis g to the power of negative 1 end exponent left parenthesis x right parenthesis right parenthesis  equals f to the power of negative 1 end exponent left parenthesis x squared plus 1 right parenthesis  equals fraction numerator negative 5 left parenthesis x squared minus 8 over denominator 4 x squared plus 2 end fraction    left parenthesis f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent right parenthesis left parenthesis 2 right parenthesis equals fraction numerator negative 5 left parenthesis negative 2 right parenthesis squared minus 8 over denominator 4 left parenthesis negative 2 right parenthesis squared plus 2 end fraction  equals fraction numerator negative 5 left parenthesis 4 right parenthesis minus 8 over denominator 4 left parenthesis 4 right parenthesis plus 2 end fraction  equals fraction numerator negative 20 minus 8 over denominator 8 plus 2 end fraction equals fraction numerator negative 28 over denominator 10 end fraction equals negative 2 comma 8

Roboguru

Diketahui  dan . Apakah ?  Sifat apa yang dapat kalian temukan dari fungsi invers pada komposisi?

Pembahasan Soal:

  • Untuk mengecek persamaan tersebut, kita cari fungsi masing-masing untuk ruas kanan dan ruas kiri.

Pada ruas kanan, invers dari fungsi linear f left parenthesis x right parenthesis dan g left parenthesis x right parenthesis adalah:

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell x plus 2 end cell row x equals cell f left parenthesis x right parenthesis minus 2 end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell x minus 2 end cell end table 

table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell 3 x minus 10 end cell row cell 3 x end cell equals cell g left parenthesis x right parenthesis plus 10 end cell row x equals cell fraction numerator g left parenthesis x right parenthesis plus 10 over denominator 3 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 plus 10 over denominator 3 end fraction end cell end table  

Sehingga,

 table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis end cell equals cell f to the power of negative 1 end exponent left parenthesis g to the power of negative 1 end exponent left parenthesis x right parenthesis right parenthesis end cell row blank equals cell f to the power of negative 1 end exponent open parentheses fraction numerator x plus 10 over denominator 3 end fraction close parentheses end cell row blank equals cell open parentheses fraction numerator x plus 10 over denominator 3 end fraction close parentheses minus 2 end cell row blank equals cell fraction numerator x plus 10 minus 6 over denominator 3 end fraction end cell row blank equals cell fraction numerator x plus 4 over denominator 3 end fraction end cell end table

Untuk invers fungsi komposisi pada ruas kiri:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses left parenthesis x right parenthesis end cell equals cell g left parenthesis f left parenthesis x right parenthesis right parenthesis end cell row cell open parentheses g ring operator f close parentheses left parenthesis x right parenthesis end cell equals cell g left parenthesis x plus 2 right parenthesis end cell row cell open parentheses g ring operator f close parentheses left parenthesis x right parenthesis end cell equals cell 3 left parenthesis x plus 2 right parenthesis minus 10 end cell row cell open parentheses g ring operator f close parentheses left parenthesis x right parenthesis end cell equals cell 3 x plus 6 minus 10 end cell row cell open parentheses g ring operator f close parentheses left parenthesis x right parenthesis end cell equals cell 3 x minus 4 end cell row cell 3 x end cell equals cell open parentheses g ring operator f close parentheses left parenthesis x right parenthesis plus 4 end cell row x equals cell fraction numerator open parentheses g ring operator f close parentheses left parenthesis x right parenthesis plus 4 over denominator 3 end fraction end cell row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator x plus 4 over denominator 3 end fraction end cell end table 

Karena ruas kanan dan ruas kiri masing-masing menghasilkan fungsi yang sama yaitu fraction numerator x plus 4 over denominator 3 end fraction, maka left parenthesis f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis equals open parentheses g ring operator f close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis.

Dengan demikian, benar bahwa left parenthesis f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis equals open parentheses g ring operator f close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis.

  • Sifat yang dapat ditemukan dari fungsi invers pada komposisi adalah invers dari suatu fungsi komposisi left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis sama dengan menginverskan masing-masing fungsi kemudian mengkomposisikannya secara terbalik sebagai mana persamaan open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals left parenthesis g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis.

Dengan demikian, sifat fungsi invers pada komposisi yang dapat ditemukan adalah open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals left parenthesis g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis.

Roboguru

Diketahui fungsi  dan ; . Invers  adalah ....

Pembahasan Soal:

Diketahui: begin mathsize 14px style f left parenthesis x right parenthesis equals 3 x plus 4 end style dan begin mathsize 14px style g left parenthesis x right parenthesis equals fraction numerator 4 x minus 5 over denominator 2 x plus 1 end fraction end style

misalkan : begin mathsize 14px style y equals open parentheses f ring operator g close parentheses open parentheses x close parentheses end style  maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell f open parentheses fraction numerator 4 x minus 5 over denominator 2 x plus 1 end fraction close parentheses end cell row blank equals cell 3 open parentheses fraction numerator 4 x minus 5 over denominator 2 x plus 1 end fraction close parentheses plus 4 end cell row blank equals cell fraction numerator 12 x minus 15 over denominator 2 x plus 1 end fraction plus fraction numerator 4 open parentheses 2 x plus 1 close parentheses over denominator 2 x plus 1 end fraction end cell row blank equals cell fraction numerator 20 x minus 11 over denominator 2 x plus 1 end fraction end cell end table end style 

maka invers dari begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator 20 x minus 11 over denominator 2 x plus 1 end fraction end cell row y equals cell fraction numerator 20 x minus 11 over denominator 2 x plus 1 end fraction end cell row cell 2 x y plus y end cell equals cell 20 x minus 11 end cell row cell y minus 11 end cell equals cell x open parentheses 20 minus 2 y close parentheses end cell row cell fraction numerator y minus 11 over denominator negative 2 y plus 20 end fraction end cell equals x end table end style  

Oleh karena  begin mathsize 14px style y equals open parentheses f ring operator g close parentheses open parentheses x close parentheses end style maka begin mathsize 14px style x equals g to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end style sehingga:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell fraction numerator y minus 11 over denominator negative 2 y plus 20 end fraction end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator x minus 11 over denominator negative 2 x plus 20 end fraction comma space x not equal to 10 end cell end table end style 

Jadi, jawaban yang tepat adalah D. 

 

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