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Diketahui  dan , maka

Pertanyaan

Diketahui f open parentheses x close parentheses equals 2 x squared plus 1 dan g open parentheses x close parentheses equals 3 x, maka open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals.... 

  1. plus-or-minus square root of fraction numerator x plus 9 over denominator 6 end fraction end root 

  2. plus-or-minus square root of fraction numerator x plus 6 over denominator 6 end fraction end root 

  3. plus-or-minus square root of fraction numerator x plus 3 over denominator 6 end fraction end root 

  4. plus-or-minus square root of fraction numerator x minus 3 over denominator 6 end fraction end root 

  5. plus-or-minus square root of fraction numerator x minus 6 over denominator 6 end fraction end root 

Pembahasan Soal:

Tentukan fungsi komposisinya.

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 2 x squared plus 1 close parentheses end cell row blank equals cell 3 open parentheses 2 x squared plus 1 close parentheses end cell row blank equals cell 6 x squared plus 3 end cell end table 

Tentukan invers dari fungsi tersebut,

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 6 x squared plus 3 end cell equals y row cell 6 x squared end cell equals cell y minus 3 end cell row cell x squared end cell equals cell fraction numerator y minus 3 over denominator 6 end fraction end cell row x equals cell plus-or-minus square root of fraction numerator y minus 3 over denominator 6 end fraction end root end cell 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 plus-or-minus square root of fraction numerator y minus 3 over denominator 6 end fraction end root 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 plus-or-minus square root of fraction numerator x minus 3 over denominator 6 end fraction end root end cell end table 

Jadi, jawaban yang tepat adalah D.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

F. Nur

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Terakhir diupdate 05 Juni 2021

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Pertanyaan yang serupa

Diketahui fungsi  dan . Invers fungsi  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.

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

Tentukan himpunan penyelesaian berikut: Diketahui dan maka

Pembahasan Soal:

Untuk menentukan begin mathsize 14px style left parenthesis f ring operator g right parenthesis end style to the power of size 14px minus size 14px 1 end exponent begin mathsize 14px style left parenthesis x right parenthesis end style pertama kita tentukan begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style.

begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses equals f open parentheses g open parentheses x close parentheses close parentheses left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals f open parentheses 2 x close parentheses left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals open parentheses 2 x close parentheses plus 4 left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals 2 x plus 4 end style

Sehingga:

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 2 x plus 4 end cell row y equals cell 2 x plus 4 end cell row cell 2 x end cell equals cell y minus 4 end cell row x equals cell fraction numerator y minus 4 over denominator 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 x minus 4 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 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 end style adalah begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x minus 4 over denominator 2 end fraction end cell end table end style

0

Roboguru

Jika =...

Pembahasan Soal:

Penyelesaian:

f left parenthesis x right parenthesis equals fraction numerator 2 x minus 1 over denominator 3 x plus 4 end fraction space d a n space g left parenthesis x right parenthesis equals fraction numerator x plus 2 over denominator 2 end fraction  left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals f left parenthesis g left parenthesis x right parenthesis right parenthesis  equals f open parentheses fraction numerator x plus 5 over denominator 2 end fraction close parentheses  equals 2 fraction numerator open parentheses begin display style fraction numerator x plus 5 over denominator 2 end fraction end style close parentheses minus 1 over denominator x open parentheses begin display style fraction numerator x plus 5 over denominator 2 end fraction end style close parentheses plus 4 end fraction  equals fraction numerator x plus 5 minus 1 over denominator begin display style fraction numerator 3 x plus 15 plus 8 over denominator 2 end fraction end style end fraction equals fraction numerator x plus 4 over denominator begin display style fraction numerator 3 x plus 23 over denominator 2 end fraction end style end fraction equals fraction numerator 2 x plus 8 over denominator 3 x plus 23 end fraction  left parenthesis f ring operator g right parenthesis 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 equals fraction numerator negative 23 plus 8 over denominator 3 x minus 2 end fraction  left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis negative 1 right parenthesis equals fraction numerator negative 23 left parenthesis negative 1 right parenthesis plus 8 over denominator 3 left parenthesis negative 1 right parenthesis minus 2 end fraction  equals fraction numerator 23 plus 8 over denominator negative 3 minus 2 end fraction equals negative 31 over 5

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Roboguru

Fungsi  dan  didefinisakan oleh  dan , Tentukan: b.

Pembahasan Soal:

Menentukan begin mathsize 14px style open parentheses g ring operator f 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 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 11 close parentheses end cell row blank equals cell 1 minus open parentheses 2 x plus 11 close parentheses end cell row blank equals cell negative 2 x minus 10 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 open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell negative 2 x minus 10 end cell row y equals cell negative 2 x minus 10 end cell row cell negative 2 x minus 10 end cell equals y row cell negative 2 x end cell equals cell y plus 10 end cell row x equals cell fraction numerator y plus 10 over denominator negative 2 end fraction end cell row x equals cell negative 1 half y minus 5 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 negative 1 half x minus 5 end cell row blank blank blank end table end style

 

0

Roboguru

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