Roboguru

Diketahui f−1(x)=5x−12x​ dan g−1(x)=7x+43​. Jika diketahui (g∘f)−1(x)=2, tentukan nilai x!

Pertanyaan

Diketahui f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 2 x over denominator 5 x minus 1 end fraction dan g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 3 over denominator 7 x plus 4 end fraction. Jika diketahui open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals 2, tentukan nilai x!

Pembahasan Soal:

Ingatlah sifat invers berikut:

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.

Berdasarkan sifat di atas, dan diketahui open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals 2, maka nilai x :

 table attributes columnalign right center left columnspacing 0px end attributes 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 2 row cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell equals 2 row cell f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell equals 2 row cell f to the power of negative 1 end exponent open parentheses fraction numerator 3 over denominator 7 x plus 4 end fraction close parentheses end cell equals 2 row cell fraction numerator 2 open parentheses begin display style fraction numerator 3 over denominator 7 x plus 4 end fraction end style close parentheses over denominator 5 open parentheses begin display style fraction numerator 3 over denominator 7 x plus 4 end fraction end style close parentheses minus 1 end fraction end cell equals 2 row cell fraction numerator begin display style fraction numerator 6 over denominator 7 x plus 4 end fraction end style over denominator begin display style fraction numerator 15 over denominator 7 x plus 4 end fraction end style minus begin display style fraction numerator 7 x plus 4 over denominator 7 x plus 4 end fraction end style end fraction end cell equals 2 row cell fraction numerator 6 over denominator 15 minus 7 x minus 4 end fraction end cell equals 2 row cell fraction numerator 6 over denominator 11 minus 7 x end fraction end cell equals 2 row cell 22 minus 14 x end cell equals 6 row cell 14 x end cell equals cell 22 minus 6 end cell row cell 14 x end cell equals 16 row x equals cell 16 over 14 end cell row x equals cell 8 over 7. end cell end table

Dengan demikian, nilai x adalah 8 over 7.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

D. Nuryani

Mahasiswa/Alumni Universitas Padjadjaran

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui fungsi f dan g adalah fungsi bijektif yang ditentukan dengan f(x)=2x+5 dan g(x)=x−2. Tentukan (g−1∘f−1)(x) dan (f−1∘g−1)(x).

Pembahasan Soal:

Fungsi komposisi merupakan penggabungan operasi dua jenis fungsi begin mathsize 14px style f left parenthesis x right parenthesis end style dan begin mathsize 14px style g left parenthesis x right parenthesis end style sehingga menghasilkan sebuah fungsi baru. Operasi fungsi komposisi biasa dilambangkan dengan "begin mathsize 14px style ring operator end style" dan dibaca komposisi atau bundaran. Fungsi baru yang dapat terbentuk dari begin mathsize 14px style f left parenthesis x right parenthesis end style dan begin mathsize 14px style g left parenthesis x right parenthesis end style adalah:

a. begin mathsize 14px style open parentheses f ring operator g close parentheses left parenthesis x right parenthesis end style artinya begin mathsize 14px style g end style dimasukkan ke size 14px f

b. begin mathsize 14px style left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end style artinya size 14px f dimasukkan ke begin mathsize 14px style g end style 

Ingat!

begin mathsize 14px style 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 equals open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style 

Tentukan begin mathsize 14px style open parentheses f ring operator g close parentheses left parenthesis x right parenthesis end style dan begin mathsize 14px style left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end style terlebih dahulu.

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 x minus 2 close parentheses end cell row blank equals cell 2 open parentheses x minus 2 close parentheses plus 5 end cell row blank equals cell 2 x minus 4 plus 5 end cell row blank equals cell 2 x plus 1 end cell 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 2 x plus 5 close parentheses end cell row blank equals cell open parentheses 2 x plus 5 close parentheses minus 2 end cell row blank equals cell 2 x plus 5 minus 2 end cell row blank equals cell 2 x plus 3 end cell end table end style 

Misalkan begin mathsize 14px style left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals m end style, maka begin mathsize 14px style 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 end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row 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 open parentheses x close parentheses end cell equals cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses comma space maka colon end cell row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell 2 x plus 1 end cell row m equals cell 2 x plus 1 end cell row cell m minus 1 end cell equals cell 2 x end cell row cell fraction numerator m minus 1 over denominator 2 end fraction end cell equals x row x equals cell fraction numerator m minus 1 over denominator 2 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 1 over denominator 2 end fraction end cell row 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 open parentheses x close parentheses end cell equals cell fraction numerator x minus 1 over denominator 2 end fraction end cell end table end style  

Misalkan begin mathsize 14px style left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis equals n end style, maka begin mathsize 14px style open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent 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 to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell equals cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell 2 x plus 3 end cell row n equals cell 2 x plus 3 end cell row cell n minus 3 end cell equals cell 2 x end cell row cell fraction numerator n minus 3 over denominator 2 end fraction end cell equals x row x equals cell fraction numerator n minus 3 over denominator 2 end fraction 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 fraction numerator x minus 3 over denominator 2 end fraction end cell row cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell equals cell fraction numerator x minus 3 over denominator 2 end fraction end cell end table end style 

Jadi, begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank 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 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x minus 1 over denominator 2 end fraction end cell end table end style dan begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x minus 3 over denominator 2 end fraction end cell end table end style.

0

Roboguru

Diketahui f(x)=x+2 dan g(x)=3x−10. Apakah (f−1∘g−1)(x)=(g∘f)−1(x)?  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.

0

Roboguru

Diketahui fungsi f(x)=−x−8 dan g(x)=32x−5​ dan h(x)=3x+4. Jika k(x)=(f−g)(x) , invers dari  (h∘k)(x) adalah ...

Pembahasan Soal:

Pertama kita tentukan fungsi begin mathsize 14px style k open parentheses x close parentheses end style dengan cara berikut

begin mathsize 14px style k open parentheses x close parentheses equals open parentheses f minus g close parentheses open parentheses x close parentheses space space space space space space equals open parentheses negative x minus 8 close parentheses minus fraction numerator 2 x minus 5 over denominator 3 end fraction space space space space space space equals fraction numerator 3 open parentheses negative x minus 8 close parentheses minus open parentheses 2 x minus 5 close parentheses over denominator 3 end fraction space space space space space space equals fraction numerator negative 3 x minus 24 minus 2 x plus 5 over denominator 3 end fraction space space space space space space equals fraction numerator negative 5 x minus 19 over denominator 3 end fraction end style 

selanjutnya akan dicari invers dari begin mathsize 14px style k open parentheses x close parentheses end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator negative 5 x minus 19 over denominator 3 end fraction end cell row cell 3 y end cell equals cell negative 5 x minus 19 end cell row cell 5 x end cell equals cell negative 19 minus 3 y end cell row x equals cell fraction numerator negative 19 minus 3 y over denominator 5 end fraction end cell row cell k to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 19 minus 3 x over denominator 5 end fraction end cell end table end style 

invers dari fungsi begin mathsize 14px style h open parentheses x close parentheses end style ditentukan dengan cara berikut

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 3 x plus 4 end cell row cell 3 x end cell equals cell y minus 4 end cell row x equals cell fraction numerator y minus 4 over denominator 3 end fraction end cell row cell h to the power of negative 1 end exponent end cell equals cell fraction numerator x minus 4 over denominator 3 end fraction end cell end table end style 

dengan sifat invers komposisi akan ditentukan begin mathsize 14px style open parentheses h ring operator k close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis h ring operator k right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell k to the power of negative 1 end exponent open parentheses h to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row cell space space space space space space space space space space space space space space space end cell equals cell k to the power of negative 1 end exponent open parentheses fraction numerator x minus 4 over denominator 3 end fraction close parentheses end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 19 minus up diagonal strike 3 begin display style fraction numerator x minus 4 over denominator up diagonal strike 3 end fraction end style over denominator 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 19 minus x plus 4 over denominator 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 15 minus x over denominator 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell negative fraction numerator x plus 15 over denominator 15 end fraction end cell end table end style

Jadi, jawaban yang tepat adalah E

0

Roboguru

2. Fungsi f:R→Rdang:R→R didefinisikan oleh f(x)=2x+11 dan g(x)=1−x. Tentukan: g. (f−1∘g−1)(x)

Pembahasan Soal:

Perhatikan penghitungan berikut!

 f open parentheses x close parentheses equals 2 x plus 11 misal colon space y equals f open parentheses x close parentheses space maka space x equals f to the power of negative 1 end exponent open parentheses x close parentheses f open parentheses x close parentheses equals 2 x plus 11 y equals 2 x plus 11 y minus 11 equals 2 x fraction numerator y minus 11 over denominator 2 end fraction equals x f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus 11 over denominator 2 end fraction  g open parentheses x close parentheses equals 1 minus x misal colon space z equals g open parentheses x close parentheses space maka space x equals g to the power of negative 1 end exponent open parentheses x close parentheses g open parentheses x close parentheses equals 1 minus x z equals 1 minus x x equals 1 minus z g to the power of negative 1 end exponent open parentheses x close parentheses equals 1 minus x  

sehingga, open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses yaitu:

 open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent open parentheses x close parentheses close parentheses open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals f to the power of negative 1 end exponent open parentheses 1 minus x close parentheses open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator open parentheses 1 minus x close parentheses minus 11 over denominator 2 end fraction open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator negative x minus 10 over denominator 2 end fraction                 

Jadi, open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator negative x minus 10 over denominator 2 end fraction .space 

0

Roboguru

Jika diketahui fungsi f(x)=4x+52x−3​dang(x)=x−1​,maka(f−1∘g−1)(−2) = ...

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

0

Roboguru

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