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Diketahui fungsi dan dengan dan , maka .

Pertanyaan

Diketahui fungsi f colon straight real numbers rightwards arrow straight real numbers dan g colon straight real numbers rightwards arrow straight real numbers dengan f open parentheses x close parentheses equals fraction numerator x plus 2 over denominator 2 x minus 3 end fraction dan g open parentheses x close parentheses equals 3 x plus 2, maka open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals horizontal ellipsis.

  1. fraction numerator 4 x minus 1 over denominator 3 x minus 6 end fraction

  2. fraction numerator 3 x minus 1 over denominator 4 x minus 6 end fraction 

  3. fraction numerator x minus 6 over denominator 3 x minus 4 end fraction 

  4. fraction numerator 6 x plus 1 over denominator 3 x minus 1 end fraction 

  5. fraction numerator negative x plus 4 over denominator 6 x minus 3 end fraction 

Pembahasan Soal:

Ingat kembali sifat komposisi fungsi 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.

Untuk menjawab soal di atas, kita perlu mencari f to the power of negative 1 end exponent open parentheses x close parentheses dan g to the power of negative 1 end exponent open parentheses x close parentheses terlebih dahulu.

Invers fungsi f open parentheses x close parentheses

Misal y equals f open parentheses x close parentheses maka  y equals fraction numerator x plus 2 over denominator 2 x minus 3 end fraction, sehingga

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

Diperoleh invers fungsi f open parentheses x close parentheses yaitu f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 3 x plus 2 over denominator 2 x minus 1 end fraction.

Invers fungsi g open parentheses x close parentheses

Misal z equals g open parentheses x close parentheses maka z equals 3 x plus 2, sehingga

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell fraction numerator z minus 2 over denominator 3 end fraction end cell end table

Diperoleh invers fungsi g open parentheses x close parentheses yaitu g to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus 2 over denominator 3 end fraction.

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

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

Jadi, open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative x plus 4 over denominator 6 x minus 3 end fraction.

Oleh karena itu, jawaban yang benar adalah E.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Fatta

Mahasiswa/Alumni Institut Pertanian Bogor

Terakhir diupdate 06 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui fungai  dan . Rumus dari  adalah ....

Pembahasan Soal:

Kita mempunyai

begin mathsize 14px style f left parenthesis x right parenthesis equals x minus 6 g left parenthesis x right parenthesis equals 2 x squared minus x plus 5 end style 

Perhatikan rumus di bawah ini!

begin mathsize 14px style 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 left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end style maka begin mathsize 14px style left parenthesis f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals left parenthesis left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end style 

Dengan demikian, kita mencari begin mathsize 14px style left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals cell g left parenthesis f left parenthesis x right parenthesis right parenthesis end cell row blank equals cell 2 left parenthesis f left parenthesis x right parenthesis right parenthesis squared minus f left parenthesis x right parenthesis plus 5 end cell row blank equals cell 2 left parenthesis x minus 6 right parenthesis squared minus left parenthesis x minus 6 right parenthesis plus 5 end cell row blank equals cell 2 left parenthesis x squared minus 12 x plus 36 right parenthesis minus x plus 6 plus 5 end cell row blank equals cell 2 x squared minus 24 x plus 72 minus x plus 11 end cell row blank equals cell 2 x squared minus 25 x plus 83 end cell end table end style

Jadi, jawaban yang tepat adalah A.

0

Roboguru

Diketahui  dan . Jika diketahui , tentukan nilai !

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.

0

Roboguru

2. Fungsi  didefinisikan oleh  dan . Tentukan: g.

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

Fungsi  dan  ditentukan oleh rumus  dan . Tentukan :

Pembahasan Soal:

Pertama kita tentukan invers dari begin mathsize 14px style straight f open parentheses straight x close parentheses equals 7 straight x minus 5 end style 
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell straight f open parentheses straight x close parentheses end cell equals cell 7 straight x minus 5 end cell row straight y equals cell 7 straight x minus 5 end cell row cell negative 7 straight x end cell equals cell negative straight y minus 5 end cell row straight x equals cell fraction numerator negative straight y minus 5 over denominator negative 7 end fraction end cell row sehingga colon blank row cell straight f to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell fraction numerator negative straight x minus 5 over denominator negative 7 end fraction end cell end table end style 
 

Selanjutnya kita tentukan invers dari  begin mathsize 14px style straight g open parentheses straight x close parentheses equals 4 straight x plus 9 end style
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell straight g open parentheses straight x close parentheses end cell equals cell 4 straight x plus 9 end cell row straight y equals cell 4 straight x plus 9 end cell row cell negative 4 straight x end cell equals cell negative straight y plus 9 end cell row straight x equals cell fraction numerator negative straight y plus 9 over denominator negative 4 end fraction end cell row sehingga colon blank row cell straight g to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell fraction numerator negative straight x plus 9 over denominator negative 4 end fraction end cell end table end style 
 

Maka begin mathsize 14px style open parentheses straight f to the power of negative 1 end exponent ring operator straight g to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses end style  sebagai berikut :
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses straight f to the power of negative 1 end exponent ring operator straight g to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses end cell equals cell fraction numerator negative open parentheses straight g to the power of negative 1 end exponent open parentheses straight x close parentheses close parentheses minus 5 over denominator negative 7 end fraction end cell row blank equals cell fraction numerator negative open parentheses begin display style fraction numerator negative straight x plus 9 over denominator negative 4 end fraction end style close parentheses minus 5 over denominator negative 7 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator straight x minus 9 over denominator negative 4 end fraction end style minus open parentheses begin display style fraction numerator negative 20 over denominator negative 4 end fraction end style close parentheses over denominator negative 7 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator straight x minus 9 plus 20 over denominator negative 4 end fraction end style over denominator negative 7 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator straight x plus 11 over denominator negative 4 end fraction end style over denominator negative 7 end fraction end cell row blank equals cell fraction numerator negative 7 open parentheses straight x plus 11 close parentheses over denominator negative 4 end fraction end cell row blank equals cell fraction numerator negative 7 straight x minus 77 over denominator negative 4 end fraction end cell end table end style 
 

Jadi, begin mathsize 14px style open parentheses straight f to the power of negative 1 end exponent ring operator straight g to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses equals fraction numerator negative 7 straight x minus 77 over denominator negative 4 end fraction end style 

0

Roboguru

Diketahui fungsi  dan . Tentukan nilai dari  adalah ...

Pembahasan Soal:

Diketahui : f left parenthesis x right parenthesis equals x minus 5g left parenthesis x right parenthesis equals 3 x plus 2 

Jika f left parenthesis x right parenthesis equals y maka f to the power of negative 1 end exponent left parenthesis y right parenthesis equals x 

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x minus 5 end cell row x equals cell y plus 5 end cell row cell f to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell y plus 5 end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell x plus 5 end cell end table 

Jika g left parenthesis x right parenthesis equals y maka g to the power of negative 1 end exponent left parenthesis y right parenthesis equals x 

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 3 x plus 2 end cell row cell 3 x end cell equals cell y minus 2 end cell row x equals cell fraction numerator y minus 2 over denominator 3 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 minus 2 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 minus 2 over denominator 3 end fraction end cell end table  

Dengan menggunakan sifat invers fungsi komposisi maka :space 

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis straight f ring operator straight g right parenthesis to the power of negative 1 end exponent left parenthesis straight x right parenthesis end cell equals cell left parenthesis straight g to the power of negative 1 end exponent ring operator straight f to the power of negative 1 end exponent right parenthesis left parenthesis straight x right parenthesis end cell row blank equals cell straight g to the power of negative 1 end exponent left parenthesis straight f to the power of negative 1 end exponent left parenthesis straight x right parenthesis right parenthesis end cell row blank equals cell straight g to the power of negative 1 end exponent left parenthesis straight x plus 5 right parenthesis end cell row blank equals cell fraction numerator left parenthesis straight x plus 5 right parenthesis minus 2 over denominator 3 end fraction end cell row blank equals cell fraction numerator straight x plus 3 over denominator 3 end fraction end cell end table 

sehinggaspace 

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

Jadi, nilai dari left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis 3 right parenthesis adalah 3.

0

Roboguru

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