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Diketahui fungsi  dan . Invers fungsi  adalah ...

Pertanyaan

Diketahui fungsi f left parenthesis x right parenthesis equals 4 x plus 1 dan g left parenthesis x right parenthesis equals fraction numerator x plus 5 over denominator 4 x minus 3 end fraction comma space x not equal to 3 over 4. Invers fungsi open parentheses f ring operator g close parentheses left parenthesis x right parenthesis adalah ...

Pembahasan Soal:

Diketahui:

f left parenthesis x right parenthesis equals 4 x plus 1
g left parenthesis x right parenthesis equals fraction numerator x plus 5 over denominator 4 x minus 3 end fraction comma space x not equal to 3 over 4

Komposisi dua fungsi open parentheses f ring operator g close parentheses 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 open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell 4 open parentheses fraction numerator x plus 5 over denominator 4 x minus 3 end fraction close parentheses plus 1 end cell row blank equals cell fraction numerator 4 x plus 20 over denominator 4 x minus 3 end fraction plus 1 end cell row blank equals cell fraction numerator 4 x plus 20 plus 4 x minus 3 over denominator 4 x minus 3 end fraction end cell row blank equals cell fraction numerator 8 x plus 17 over denominator 4 x minus 3 end fraction end cell end table 

Invers fungsi open parentheses f ring operator g close parentheses left parenthesis x right parenthesis dengan mengubah ke variabel y telebih dahulu.

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 fraction numerator 8 x plus 17 over denominator 4 x minus 3 end fraction end cell row y equals cell fraction numerator 8 x plus 17 over denominator 4 x minus 3 end fraction end cell row cell 4 x y minus 3 y end cell equals cell 8 x plus 17 end cell row cell 4 x y minus 8 x end cell equals cell 3 y plus 17 end cell row cell x open parentheses 4 y minus 8 close parentheses end cell equals cell 3 y plus 17 end cell row x equals cell fraction numerator 3 y plus 17 over denominator 4 y minus 8 end fraction end cell end table

Maka inversnya adalah:

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell fraction numerator 3 y plus 17 over denominator 4 y minus 8 end fraction end cell row cell y apostrophe end cell equals cell fraction numerator 3 x plus 17 over denominator 4 x minus 8 end fraction end cell row cell open parentheses f ring operator g close parentheses apostrophe open parentheses x close parentheses end cell equals cell fraction numerator 3 x plus 17 over denominator 4 x minus 8 end fraction end cell end table 

Jadi, invers fungsi open parentheses f ring operator g close parentheses left parenthesis x right parenthesis adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f ring operator g close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe 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 3 x plus 17 over denominator 4 x minus 8 end fraction end cell end table.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Dwi

Mahasiswa/Alumni Universitas Sriwijaya

Terakhir diupdate 06 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui fungsi  dan  adalah fungsi bijektif yang ditentukan dengan  dan . Tentukan  dan .

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  dan , maka

Pembahasan Soal:

Fungsi Komposisi

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

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

b. begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses end style artinya begin mathsize 14px style f end style dimasukkan ke size 14px g 

Fungsi Invers

Fungsi invers atau fungsi kebalikan merupakan suatu fungsi yang berkebalikan dari fungsi asalnya. Suatu fungsi f memiliki fungsi invers (kebalikan) f to the power of negative 1 end exponent jika f merupakan fungsi satu-satu dan fungsi pada (bijektif).

space space space space f colon space x rightwards arrow y f to the power of negative 1 end exponent colon space y rightwards arrow x 

Ada 3 langkah untuk menentukan fungsi invers.

  • Ubahlah bentuk y equals f open parentheses x close parentheses menjadi bentuk x equals f open parentheses y close parentheses.
  • Tuliskan x sebagai f to the power of negative 1 end exponent open parentheses y close parentheses sehingga f to the power of negative 1 end exponent open parentheses y close parentheses equals f open parentheses y close parentheses.
  • Ubahlah variabel y dengan x sehingga diperoleh rumus fungsi f to the power of negative 1 end exponent open parentheses x close parentheses.

Diketahui  f open parentheses x close parentheses equals x minus 2 dan g open parentheses x close parentheses equals 3 x plus 1, maka open parentheses f ring operator g close parentheses 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 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 plus 1 close parentheses end cell row blank equals cell 3 x plus 1 minus 2 end cell row blank equals cell 3 x minus 1 end cell end table 

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

Jadi, jawaban yang tepat adalah C.

0

Roboguru

Diketahui fungsi  dan  dengan  dan , maka

Pembahasan Soal:

Diketahui fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 3 x plus 5 over denominator x end fraction end style dan begin mathsize 14px style g left parenthesis x right parenthesis equals 2 x minus 1 end style. Komposisi fungsi undefined dapat dihitung seperti berikut,

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 f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell row blank equals cell f left parenthesis 2 x minus 1 right parenthesis end cell row blank equals cell fraction numerator 3 left parenthesis 2 x minus 1 right parenthesis plus 5 over denominator 2 x minus 1 end fraction end cell row blank equals cell fraction numerator 6 x minus 3 plus 5 over denominator 2 x minus 1 end fraction end cell row blank equals cell fraction numerator 6 x plus 2 over denominator 2 x minus 1 end fraction comma space x not equal to 1 half end cell end table end style 

Bentuk umum invers fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction end style adalah

begin mathsize 14px style f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative d x plus b over denominator c x minus a end fraction comma space x not equal to a over c end style.

Berdasarkan rumus invers di atas, bentuk invers untuk fungsi begin mathsize 14px style open parentheses f ring operator g close parentheses left parenthesis x right parenthesis equals fraction numerator 6 x plus 2 over denominator 2 x minus 1 end fraction end style adalah

begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative left parenthesis negative 1 right parenthesis x plus 2 over denominator 2 x minus 6 end fraction equals fraction numerator x plus 2 over denominator 2 x minus 6 end fraction comma space x not equal to 3 end style 

Dengan demikian, fungsi invers begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis end style adalah fraction numerator size 14px x size 14px plus size 14px 2 over denominator size 14px 2 size 14px x size 14px minus size 14px 6 end fraction.

Jadi, jawaban yang tepat adalah E. 

0

Roboguru

Jika  dan , maka  yang tepat adalah ....

Pembahasan Soal:

Cara 1

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

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f ring operator h close parentheses open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses h open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell g open parentheses f open parentheses 4 x minus 1 close parentheses close parentheses end cell row blank equals cell g open parentheses 2 open parentheses 4 x minus 1 close parentheses plus 2 close parentheses end cell row blank equals cell g open parentheses 8 x minus 2 plus 2 close parentheses end cell row blank equals cell g open parentheses 8 x close parentheses end cell row blank equals cell 3 minus 8 x end cell end table end style 

 

Kemudian, tentukan begin mathsize 14px style left parenthesis g ring operator f ring operator h right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end style

Jika open parentheses g ring operator f ring operator h 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 left parenthesis g ring operator f ring operator h right parenthesis left parenthesis x right parenthesis end cell equals cell 3 minus 8 x end cell row y equals cell 3 minus 8 x end cell row cell 8 x end cell equals cell 3 minus y end cell row x equals cell fraction numerator 3 minus y over denominator 8 end fraction end cell row cell left parenthesis g ring operator f ring operator h right parenthesis to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell fraction numerator 3 minus y over denominator 8 end fraction end cell end table end style 

Dengan demikian, begin mathsize 14px style open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 3 minus x over denominator 8 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 comma space g to the power of negative 1 end exponent comma end style dan begin mathsize 14px style h to the power of negative 1 end exponent end style terlebih dahulu.

Dimulai dari mencari f to the power of negative 1 end exponent. Perhatikan bahwa

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

Kemudian, akan dicari g to the power of negative 1 end exponent. Perhatikan bahwa

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

Selanjutnya, akan dicari h to the power of negative 1 end exponent. Perhatikan bahwa

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

Ingat kembali bahwa begin mathsize 14px style open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses h to the power of negative 1 end exponent ring operator 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, maka 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 ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses h to the power of negative 1 end exponent ring operator 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 h to the power of negative 1 end exponent open parentheses 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 close parentheses end cell row blank equals cell h to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses 3 minus x close parentheses close parentheses end cell row blank equals cell h to the power of negative 1 end exponent open parentheses fraction numerator open parentheses 3 minus x close parentheses minus 2 over denominator 2 end fraction close parentheses end cell row blank equals cell h to the power of negative 1 end exponent open parentheses fraction numerator 1 minus x over denominator 2 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses fraction numerator 1 minus x over denominator 2 end fraction close parentheses plus 1 over denominator 4 end fraction end cell row blank equals cell fraction numerator fraction numerator 1 minus x over denominator 2 end fraction plus 2 over 2 over denominator 4 end fraction end cell row blank equals cell fraction numerator fraction numerator 3 minus x over denominator 2 end fraction over denominator 4 end fraction end cell row blank equals cell fraction numerator 3 minus x over denominator 8 end fraction end cell end table end style 

Jadi, jawaban yang tepat adalah E.

1

Roboguru

Diketahui fungsi f dan h, dengan  untuk setiap bilangan x real. Untuk , maka adalah...

Pembahasan Soal:

f left parenthesis x right parenthesis equals 10 x space d a n space h left parenthesis x right parenthesis equals x squared plus 2  f to the power of negative 1 end exponent left parenthesis x right parenthesis equals log space x  h left parenthesis x right parenthesis equals x squared plus 2  h left parenthesis x squared right parenthesis equals left parenthesis x squared right parenthesis squared plus 2 equals x squared plus 2  h left parenthesis x squared right parenthesis minus 2 equals x 4 plus 2 minus 2 equals x squared  f to the power of negative 1 end exponent left parenthesis x right parenthesis equals f to the power of negative 1 end exponent left square bracket h left parenthesis x squared right parenthesis minus 2 right square bracket equals f to the power of negative 1 end exponent left parenthesis x to the power of 4 right parenthesis equals log x to the power of 4

0

Roboguru

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