Roboguru

Diketahui  dan . Tentukan fungsi invers: , , dan

Pertanyaan

Diketahui f left parenthesis x right parenthesis equals x plus 2 dan g left parenthesis x right parenthesis equals 3 x minus 10.

  • Tentukan fungsi invers: f left parenthesis x right parenthesisg left parenthesis x right parenthesis, dan left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis 

Pembahasan Soal:

Invers dari fungsi linear f left parenthesis x right parenthesis adalah sebagai berikut:

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 

Dengan demikian, fungsi invers dari f left parenthesis x right parenthesis adalah f to the power of negative 1 end exponent left parenthesis x right parenthesis equals x minus 2.

Invers dari fungsi linear g left parenthesis x right parenthesis adalah sebagai berikut:

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 )+ 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  

Dengan demikian, fungsi invers dari g left parenthesis x right parenthesis adalah g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x plus 10 over denominator 3 end fraction.

Untuk menentukan fungsi invers komposisi dari left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis, maka kita perlu cari terlebih dahulu left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses left parenthesis x right parenthesis end cell equals cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell row blank equals cell f left parenthesis 3 x minus 10 right parenthesis end cell row blank equals cell left parenthesis 3 x minus 10 right parenthesis plus 2 end cell row blank equals cell 3 x minus 8 end cell end table 

Maka invers dari fungsi komposisi left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis yaitu sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell 3 x minus 8 end cell row cell 3 x end cell equals cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis plus 8 end cell row x equals cell fraction numerator left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis plus 8 over denominator 3 end fraction end cell row cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator x plus 8 over denominator 3 end fraction end cell end table 

Dengan demikian, fungsi invers dari fungsi komposisi left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis adalah left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x plus 8 over denominator 3 end fraction.

Jadi, fungdi invers dari f left parenthesis x right parenthesisg left parenthesis x right parenthesis, dan left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis yaitu f to the power of negative 1 end exponent left parenthesis x right parenthesis equals x minus 2g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x plus 10 over denominator 3 end fraction, dan left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x plus 8 over denominator 3 end fraction.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Fathurachman

Mahasiswa/Alumni Universitas Pertamina

Terakhir diupdate 02 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui, tentukan : b.

Pembahasan Soal:

begin mathsize 14px style table row cell open parentheses f o 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 equals cell f open parentheses 6 x minus 2 close parentheses end cell blank blank row blank blank blank equals cell fraction numerator 5 open parentheses 6 x minus 2 close parentheses minus 3 over denominator 6 x minus 2 plus 2 end fraction end cell blank blank row blank blank blank equals cell fraction numerator 30 x minus 13 over denominator 6 x end fraction end cell blank blank row cell open parentheses f o g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 13 over denominator 6 x minus 30 end fraction end cell blank blank blank blank row cell g open parentheses x close parentheses to the power of negative 1 end exponent end cell equals cell fraction numerator x plus 2 over denominator 6 end fraction end cell atau cell 1 over 6 open parentheses x plus 2 close parentheses end cell blank blank end table end style

Roboguru

Diketahui  dan . a. Tentukan fungsi invers : .

Pembahasan Soal:

Dengan menggunakan konsep fungsi komposisi diperoleh

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

Dengan demikian f to the power of negative 1 end exponent open parentheses x close parentheses equals x minus 2.

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

Dengan demikian Error converting from MathML to accessible text..

Dengan menggunakan konsep komposisi fungsi diperoleh

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 3 x minus 10 close parentheses end cell row blank equals cell 3 x minus 10 plus 2 end cell row blank equals cell 3 x minus 8 end cell row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals y row cell 3 x minus 8 end cell equals y row cell 3 x end cell equals cell y plus 8 end cell row blank blank blank row x equals cell fraction numerator y plus 8 over denominator 3 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 plus 8 over denominator 3 end fraction end cell end table

Dengan demikian Error converting from MathML to accessible text..

Roboguru

Diketahui  dan , tentukan: a.  dan   b.  dan

Pembahasan Soal:

a. f open parentheses x close parentheses equals fraction numerator 5 x minus 3 over denominator x plus 2 end fraction comma space x not equal to negative 2, inversnya adalah

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator 5 x minus 3 over denominator x plus 2 end fraction end cell row cell x y plus 2 y end cell equals cell 5 x minus 3 end cell row cell x y minus 5 x end cell equals cell negative 2 y minus 3 end cell row cell x open parentheses y minus 5 close parentheses end cell equals cell negative 2 y minus 3 end cell row x equals cell fraction numerator negative 2 y minus 3 over denominator y minus 5 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 2 x minus 3 over denominator x minus 5 end fraction comma space x not equal to 5 end cell end table

dan, g open parentheses x close parentheses equals 6 x minus 2, maka inversnya adalah

y equals 6 x minus 2 6 x equals y plus 2 x equals fraction numerator y plus 2 over denominator 6 end fraction g to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x plus 2 over denominator 6 end fraction

b. Invers dari fungsi komposisinya adalah sebagai berikut

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 negative 2 x minus 3 over denominator x minus 5 end fraction close parentheses end cell row blank equals cell fraction numerator fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction plus 2 over denominator 6 end fraction end cell row blank equals cell fraction numerator fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction plus 2 over denominator 6 end fraction cross times fraction numerator x minus 5 over denominator x minus 5 end fraction end cell row blank equals cell fraction numerator negative 2 x minus 3 plus 2 x minus 10 over denominator 6 x minus 30 end fraction end cell row blank equals cell negative fraction numerator 13 over denominator 6 x minus 30 end fraction comma space x not equal to 5 end cell end table

Dan, 

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 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 row blank equals 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 row blank equals cell f to the power of negative 1 end exponent open parentheses fraction numerator x plus 2 over denominator 6 end fraction close parentheses end cell row blank equals cell fraction numerator negative 2 open parentheses fraction numerator x plus 2 over denominator 6 end fraction close parentheses minus 3 over denominator fraction numerator x plus 2 over denominator 6 end fraction minus 5 end fraction end cell row blank equals cell fraction numerator negative 2 open parentheses fraction numerator x plus 2 over denominator 6 end fraction close parentheses minus 3 over denominator fraction numerator x plus 2 over denominator 6 end fraction minus 5 end fraction cross times 6 over 6 end cell row blank equals cell fraction numerator negative 2 x minus 4 minus 18 over denominator x plus 2 minus 30 end fraction end cell row blank equals cell fraction numerator negative 2 x minus 22 over denominator x minus 28 end fraction comma space x not equal to 28 end cell end table 

 

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

Fungsi  dan  didefinisakan oleh  dan , Tentukan: g.

Pembahasan Soal:

Menentukan begin mathsize 14px style f 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 f left parenthesis x right parenthesis end cell equals cell 2 x plus 11 end cell row y equals cell 2 x plus 11 end cell row cell 2 x plus 11 end cell equals y row cell 2 x end cell equals cell y minus 11 end cell row x equals cell fraction numerator y minus 11 over denominator 2 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator x minus 11 over denominator 2 end fraction end cell row blank blank blank end table end style  

Menentukan begin mathsize 14px style g 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 g left parenthesis x right parenthesis end cell equals cell 1 minus x end cell row y equals cell 1 minus x end cell row cell 1 minus x end cell equals y row cell negative x end cell equals cell y minus 1 end cell row x equals cell 1 minus y end cell row cell g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell 1 minus x end cell end table end style

  Menentukan 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 end style

begin mathsize 14px style 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 open parentheses x close parentheses right parenthesis end cell row blank equals cell f to the power of negative 1 end exponent left parenthesis 1 minus x right parenthesis end cell row blank equals cell fraction numerator left parenthesis 1 minus x right parenthesis minus 11 over denominator 2 end fraction end cell row blank equals cell fraction numerator negative x minus 10 over denominator 2 end fraction end cell row blank equals cell negative 1 half x minus 5 end cell end table end style  

Dengan demikian, hasil dari (f1g1)(x)=21x5.

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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