Roboguru

Diketahui , ,  berturut-turut menyatakan invers fungsi f, g, dan h. Jika  dan , nilai  adalah...

Pertanyaan

Diketahui f to the power of negative 1 end exponentg to the power of negative 1 end exponenth to the power of negative 1 end exponent berturut-turut menyatakan invers fungsi f, g, dan h. Jika open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals 2 x minus 4 dan open parentheses h ring operator g close parentheses open parentheses x close parentheses equals fraction numerator x minus 3 over denominator 2 x plus 1 end fraction comma space x not equal to 1 half, nilai f left parenthesis 8 right parenthesis adalah...space

  1. negative 3 over 11

  2. negative 4 over 5 space

  3. negative 9 over 11 space

  4. negative 12 over 11 space

  5. negative 5 over 11

Pembahasan Soal:

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 ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell equals cell 2 x minus 4 end cell row cell open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell 2 x minus 4 end cell row cell open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses end cell equals cell fraction numerator x plus 4 over denominator 2 end fraction end cell row blank equals cell 1 half x plus 2 end cell row blank blank blank row cell open parentheses h ring operator g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator x minus 3 over denominator 2 x plus 1 end fraction end cell row cell open parentheses h ring operator g close parentheses open parentheses f open parentheses x close parentheses close parentheses end cell equals cell fraction numerator f open parentheses x close parentheses minus 3 over denominator 2 f left parenthesis x right parenthesis plus 1 end fraction end cell row cell h ring operator g ring operator f open parentheses x close parentheses end cell equals cell fraction numerator f open parentheses x close parentheses minus 3 over denominator 2 f left parenthesis x right parenthesis plus 1 end fraction end cell row blank equals cell 1 half x plus 2 end cell row blank blank blank row cell f left parenthesis x right parenthesis minus 3 end cell equals cell x f open parentheses x close parentheses plus 4 f open parentheses x close parentheses plus 1 half open parentheses x close parentheses plus 2 end cell row cell 2 f left parenthesis x right parenthesis minus 6 end cell equals cell 2 x f left parenthesis x right parenthesis plus 8 f left parenthesis x right parenthesis plus x plus 4 end cell row blank blank blank row cell 2 x left parenthesis f x right parenthesis plus 6 f left parenthesis x right parenthesis plus x plus 10 end cell equals 0 row cell f left parenthesis x right parenthesis left parenthesis 2 x plus 6 right parenthesis plus x plus 10 end cell equals 0 row cell f left parenthesis x right parenthesis end cell equals cell negative fraction numerator x plus 10 over denominator 2 x plus 6 end fraction end cell row cell f left parenthesis 8 right parenthesis end cell equals cell negative fraction numerator 8 plus 10 over denominator 2 left parenthesis 8 right parenthesis plus 6 end fraction end cell row blank equals cell negative 18 over 22 end cell row blank equals cell negative 9 over 11 end cell end table

Oleh karena itu, jawaban yang benar adalah C.
 

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 f(x) = 4x + 2 dan . Invers dari (g o f)(x) adalah ....

Pembahasan Soal:

begin mathsize 14px style open parentheses straight g ring operator straight f close parentheses open parentheses straight x close parentheses equals straight g open parentheses straight f open parentheses straight x close parentheses close parentheses  equals straight g open parentheses 4 straight x plus 2 close parentheses  equals fraction numerator left parenthesis 4 straight x plus 2 right parenthesis minus 3 over denominator left parenthesis 4 straight x plus 2 right parenthesis plus 1 end fraction  equals fraction numerator 4 straight x minus 1 over denominator 4 straight x plus 3 end fraction    Invers space dari space straight h left parenthesis straight x right parenthesis equals fraction numerator ax plus straight b over denominator cx plus straight d end fraction space adalah space straight h to the power of negative 1 end exponent left parenthesis straight x right parenthesis equals fraction numerator negative dx plus straight b over denominator cx minus straight a end fraction  Sehingga space invers space dari space left parenthesis straight g ring operator straight f right parenthesis left parenthesis straight x right parenthesis equals fraction numerator 4 straight x minus 1 over denominator 4 straight x plus 3 end fraction space adalah  open parentheses straight g ring operator straight f close parentheses to the power of negative 1 end exponent open parentheses straight x close parentheses equals fraction numerator negative 3 straight x minus 1 over denominator 4 straight x minus 4 end fraction  open parentheses straight g ring operator straight f close parentheses to the power of negative 1 end exponent open parentheses straight x close parentheses equals fraction numerator 3 straight x plus 1 over denominator 4 minus 4 straight x end fraction end style

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.

Roboguru

Tentukan .

Pembahasan Soal:

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 space space space space space space space space space space space space space equals f open parentheses fraction numerator 11 x plus 4 over denominator x minus 5 end fraction close parentheses space space space space space space space space space space space space space equals fraction numerator 11 x plus 4 over denominator x minus 5 end fraction minus 4 space space space space space space space space space space space space space equals fraction numerator open parentheses 11 x plus 4 close parentheses minus 4 open parentheses x minus 5 close parentheses over denominator x minus 5 end fraction space space space space space space space space space space space space space equals fraction numerator 7 x plus 24 over denominator x minus 5 end fraction 

sekarang kita akan mencari invers fungsinya,

misal space fraction numerator 7 x plus 24 over denominator x minus 5 end fraction equals a space space space space space space space space space space 7 x plus 24 equals a open parentheses x minus 5 close parentheses space space space space space space space space space space 7 x plus 24 equals a x minus 5 a space space space space space space space space space space 7 x minus a x equals negative 5 a minus 24 space space space space space space space space space space open parentheses 7 minus a close parentheses x equals negative 5 a minus 24 space space space space space space space space space space space space space space space space space space space space x equals fraction numerator negative 5 a minus 24 over denominator 7 minus a end fraction space space space space space space space space space space space space space space space space space space space space x equals fraction numerator 5 a plus 24 over denominator a minus 7 end fraction space space space space space space space space space space space space rightwards arrow space space space space space open parentheses f ring operator g close parentheses to the power of negative 1 end exponent equals fraction numerator 5 x plus 24 over denominator x minus 7 end fraction 

Jadi, open parentheses f ring operator g close parentheses to the power of negative 1 end exponent equals fraction numerator 5 x plus 24 over denominator x minus 7 end fraction.

Roboguru

Diketahui fungsi  dan  dan  Jika  , invers dari   adalah ...

Pembahasan Soal:

Pertama kita tentukan fungsi begin mathsize 14px style k open parentheses x close parentheses end style dengan cara berikut

begin mathsize 14px style k open parentheses x close parentheses equals open parentheses f minus g close parentheses open parentheses x close parentheses space space space space space space equals open parentheses negative x minus 8 close parentheses minus fraction numerator 2 x minus 5 over denominator 3 end fraction space space space space space space equals fraction numerator 3 open parentheses negative x minus 8 close parentheses minus open parentheses 2 x minus 5 close parentheses over denominator 3 end fraction space space space space space space equals fraction numerator negative 3 x minus 24 minus 2 x plus 5 over denominator 3 end fraction space space space space space space equals fraction numerator negative 5 x minus 19 over denominator 3 end fraction end style 

selanjutnya akan dicari invers dari begin mathsize 14px style k open parentheses x close parentheses end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator negative 5 x minus 19 over denominator 3 end fraction end cell row cell 3 y end cell equals cell negative 5 x minus 19 end cell row cell 5 x end cell equals cell negative 19 minus 3 y end cell row x equals cell fraction numerator negative 19 minus 3 y over denominator 5 end fraction end cell row cell k to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 19 minus 3 x over denominator 5 end fraction end cell end table end style 

invers dari fungsi begin mathsize 14px style h open parentheses x close parentheses end style ditentukan dengan cara berikut

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 3 x plus 4 end cell row cell 3 x end cell equals cell y minus 4 end cell row x equals cell fraction numerator y minus 4 over denominator 3 end fraction end cell row cell h to the power of negative 1 end exponent end cell equals cell fraction numerator x minus 4 over denominator 3 end fraction end cell end table end style 

dengan sifat invers komposisi akan ditentukan begin mathsize 14px style open parentheses h ring operator k close parentheses 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 left parenthesis h ring operator k right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell k to the power of negative 1 end exponent open parentheses h to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row cell space space space space space space space space space space space space space space space end cell equals cell k to the power of negative 1 end exponent open parentheses fraction numerator x minus 4 over denominator 3 end fraction close parentheses end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 19 minus up diagonal strike 3 begin display style fraction numerator x minus 4 over denominator up diagonal strike 3 end fraction end style over denominator 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 19 minus x plus 4 over denominator 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 15 minus x over denominator 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell negative fraction numerator x plus 15 over denominator 15 end fraction end cell end table end style

Jadi, jawaban yang tepat adalah E

Roboguru

Jika fungsi  dan  memiliki invers dan memenuhi , rumus invers fungsi  adalah ....

Pembahasan Soal:

Diketahui: begin mathsize 14px style f left parenthesis 2 x right parenthesis equals g left parenthesis x minus 5 right parenthesis end style maka 

begin mathsize 14px style 2 x equals f to the power of negative 1 end exponent times g open parentheses x minus 5 close parentheses end style 

Misalkan begin mathsize 14px style g open parentheses x minus 5 close parentheses equals a end style maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses x minus 5 close parentheses end cell equals a row cell x minus 5 end cell equals cell g to the power of negative 1 end exponent open parentheses a close parentheses end cell row x equals cell g to the power of negative 1 end exponent open parentheses a close parentheses plus 5 end cell end table end style 

Sehingga diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 open parentheses g to the power of negative 1 end exponent open parentheses a close parentheses plus 5 close parentheses end cell equals cell f to the power of negative 1 end exponent open parentheses a close parentheses end cell row cell 2 times g to the power of negative 1 end exponent open parentheses a close parentheses plus 10 end cell equals cell f to the power of negative 1 end exponent open parentheses a close parentheses end cell row cell 2 times g to the power of negative 1 end exponent open parentheses x close parentheses plus 10 end cell equals cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell end table end style 

Jadi, jawaban yang tepat adalah E.

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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