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Jika dan , , nilai dari adalah....

Pertanyaan

Jika f open parentheses x close parentheses equals x space plus space 2 spacedan g left parenthesis x right parenthesis equals fraction numerator x over denominator x plus 5 end fractionx not equal to 5, nilai dariopen parentheses g to the power of negative 1 end exponent ring operator f close parentheses open parentheses 4 close parentheses adalah....space

  1. negative 8 space

  2. negative 6 space

  3. negative 2 space

  4. 4

  5. 6 space

Pembahasan Soal:

Invers fungsi g open parentheses x close parentheses

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

Maka :

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 close parentheses open parentheses x close parentheses end cell equals cell g to the power of negative 1 end exponent open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses x plus 2 close parentheses end cell row blank equals cell fraction numerator negative 5 open parentheses x plus 2 close parentheses over denominator open parentheses x plus 2 minus 1 close parentheses end fraction end cell row blank equals cell fraction numerator negative 5 x minus 10 over denominator x plus 1 end fraction end cell row cell open parentheses g to the power of negative 1 end exponent ring operator f close parentheses open parentheses 4 close parentheses end cell equals cell fraction numerator negative 5 left parenthesis 4 right parenthesis minus 10 over denominator 4 plus 1 end fraction end cell row blank equals cell fraction numerator negative 20 minus 10 over denominator 5 end fraction end cell row blank equals cell negative 30 over 5 end cell row blank equals cell negative 6 end cell end table

Oleh karena itu, jawaban yang benar adalah B.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

G. Albiah

Mahasiswa/Alumni Universitas Galuh Ciamis

Terakhir diupdate 01 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

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: h.

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 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 end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 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 end cell 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 x minus 11 over denominator 2 end fraction close parentheses end cell row blank equals cell 1 minus open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses end cell row blank equals cell 2 over 2 minus open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses end cell row blank equals cell fraction numerator 2 minus x plus 11 over denominator 2 end fraction end cell row blank equals cell fraction numerator 13 minus x over denominator 2 end fraction end cell end table end style  

Roboguru

Diketahui fungsi . Jika invers fungsi  dan invers fungsi . Tentukan:

Pembahasan Soal:

Diketahui begin mathsize 14px style f left parenthesis x right parenthesis equals 2 x minus 1 space dan space g left parenthesis x right parenthesis equals fraction numerator x minus 4 over denominator 6 x end fraction comma x not equal to 0 end style

Menentukan begin mathsize 14px style left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell row blank equals cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell row blank equals cell f left parenthesis fraction numerator x minus 4 over denominator 6 x end fraction right parenthesis end cell row blank equals cell 2 left parenthesis fraction numerator x minus 4 over denominator 6 x end fraction right parenthesis minus 1 end cell row blank equals cell fraction numerator 2 x minus 8 over denominator 6 x end fraction minus fraction numerator 6 x over denominator 6 x end fraction end cell row blank equals cell fraction numerator 2 x minus 8 minus 6 x over denominator 6 x end fraction end cell row blank equals cell fraction numerator negative 4 x minus 8 over denominator 6 x end fraction end cell row blank equals cell fraction numerator 2 left parenthesis negative 2 x minus 4 right parenthesis over denominator 6 x end fraction end cell row blank equals cell fraction numerator negative 2 x minus 4 over denominator 3 x end fraction comma x not equal to 0 end cell end table end style 

Menentukan  begin mathsize 14px style left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent 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 ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator negative 2 x minus 4 over denominator 3 x end fraction end cell row y equals cell fraction numerator negative 2 x minus 4 over denominator 3 x end fraction end cell row cell 3 x y end cell equals cell negative 2 x minus 4 end cell row cell 3 x y plus 2 x end cell equals cell negative 4 end cell row cell x left parenthesis 3 y plus 2 right parenthesis end cell equals cell negative 4 end cell row x equals cell fraction numerator negative 4 over denominator 3 y plus 2 end fraction end cell row cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell fraction numerator negative 4 over denominator 3 y plus 2 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 negative 4 over denominator 3 x plus 2 end fraction comma x not equal to fraction numerator negative 2 over denominator 3 end fraction end cell end table end style 

a. Jadi, begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis 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 negative 4 over denominator 3 x plus 2 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank not equal to blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 2 over denominator 3 end fraction end cell end table end style.

Menentukan 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 blank blank cell left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end cell row blank equals cell g left parenthesis f left parenthesis x right parenthesis right parenthesis end cell row blank equals cell g left parenthesis 2 x minus 1 right parenthesis end cell row blank equals cell fraction numerator 2 x minus 1 minus 4 over denominator 6 left parenthesis 2 x minus 1 right parenthesis end fraction end cell row blank equals cell fraction numerator 2 x minus 5 over denominator 12 x minus 6 end fraction comma space x not equal to 1 half end cell end table end style 

Menentukan begin mathsize 14px style left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent 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 fraction numerator 2 x minus 5 over denominator 12 x minus 6 end fraction end cell row y equals cell fraction numerator 2 x minus 5 over denominator 12 x minus 6 end fraction end cell row cell y left parenthesis 12 x minus 6 right parenthesis end cell equals cell 2 x minus 5 end cell row cell 12 x y minus 6 y end cell equals cell 2 x minus 5 end cell row cell 12 x y minus 2 x end cell equals cell 6 y minus 5 end cell row cell x left parenthesis 12 y minus 2 right parenthesis end cell equals cell 6 y minus 5 end cell row x equals cell fraction numerator 6 y minus 5 over denominator 12 y minus 2 end fraction end cell row cell left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell fraction numerator 6 y minus 5 over denominator 12 y minus 2 end fraction end cell row cell left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 6 x minus 5 over denominator 12 x minus 2 end fraction comma space x not equal to 1 over 6 end cell row blank blank blank row blank blank blank end table end style 

b. Jadi, begin mathsize 14px style left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 6 x minus 5 over denominator 12 x minus 2 end fraction comma x not equal to 1 over 6 end style

 

Roboguru

Jika  dan , maka  yang tepat adalah ....

Pembahasan Soal:

Cara 1:

Pertama kita cari fungsi komposisi begin mathsize 14px style g ring operator f end style.

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

Selanjutnya, kita cari invers dari begin mathsize 14px style g ring operator f end style.

Perhatikan jika open parentheses g ring operator f close parentheses open parentheses x close parentheses equals y, maka didapat perhitungan sebagai berikut.

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 2 x plus 2 end cell row y equals cell 2 x plus 2 end cell row cell y minus 2 end cell equals cell 2 x end cell row cell fraction numerator y minus 2 over denominator 2 end fraction end cell equals x end table end style 

Dengan demikian, kita peroleh hasil sebagai berikut.

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 y 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 x 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 fraction numerator y minus 2 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 2 over denominator 2 end fraction end cell end table end style 

 

Cara 2:

Kita cari masing-masing begin mathsize 14px style f to the power of negative 1 end exponent end style dan begin mathsize 14px style g to the power of negative 1 end exponent end style terlebih dahulu.

Dimulai dari mencari begin mathsize 14px style f to the power of negative 1 end exponent end style, perhatikan bahwa

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

Kemudian, akan dicari begin mathsize 14px style g to the power of negative 1 end exponent end style. Perhatikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses x close parentheses end cell equals cell 2 x plus 4 end cell row y equals cell 2 x plus 4 end cell row cell y minus 4 end cell equals cell 2 x end cell row cell fraction numerator y minus 4 over denominator 2 end fraction end cell equals x end table 

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

Akibatnya, didapat hasil sebagai berikut.

begin mathsize 14px style 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 minus 4 over denominator 2 end fraction close parentheses end cell row blank equals cell fraction numerator x minus 4 over denominator 2 end fraction plus 1 end cell row blank equals cell fraction numerator x minus 4 over denominator 2 end fraction plus 2 over 2 end cell row blank equals cell fraction numerator x minus 2 over denominator 2 end fraction end cell end table end style 

Jadi, jawaban yang tepat adalah C.

Roboguru

Diketahui , , dan . Jika , nilai dari

Pembahasan Soal:

Fungsi komposisi tiga fungsi tersebut adalah sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g ring operator h close parentheses open parentheses x close parentheses end cell equals cell open parentheses f ring operator open parentheses g ring operator h close parentheses close parentheses open parentheses x close parentheses end cell row blank equals cell f open parentheses g open parentheses h open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell f open parentheses 4 minus 3 open parentheses 2 x plus 3 close parentheses close parentheses end cell row blank equals cell f open parentheses negative 6 x minus 5 close parentheses end cell row blank equals cell open parentheses negative 6 x minus 5 close parentheses plus 2 end cell row blank equals cell negative 6 x minus 3 end cell end table end style

Invers dari fungsi komposisi tersebut adalah sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative 6 x minus 3 end cell row cell 6 x end cell equals cell negative y minus 3 end cell row x equals cell fraction numerator negative y minus 3 over denominator 6 end fraction end cell row cell open parentheses f ring operator g ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative x minus 3 over denominator 6 end fraction end cell end table end style

Jika begin mathsize 14px style open parentheses f ring operator g ring operator h close parentheses to the power of negative 1 end exponent open parentheses text a end text close parentheses equals 1 end style, maka nilai begin mathsize 14px style text a end text end style dapat ditentukan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g ring operator h close parentheses to the power of negative 1 end exponent open parentheses text a end text close parentheses end cell equals 1 row cell fraction numerator negative text a end text minus 3 over denominator 6 end fraction end cell equals 1 row cell negative text a end text minus 3 end cell equals 6 row cell negative text a end text end cell equals 9 row cell text a end text end cell equals cell negative 9 end cell end table end style

Oleh karena itu, jawaban yang tepat adalah A.

 

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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