Roboguru

Jika f:R→R dan g:R→R dengan f(x)=2x​ dan g(x)=x2−1. NIlai dari (f∘g)−1(23​​) adalah....

Pertanyaan

Jika begin mathsize 14px style f colon R rightwards arrow R end style dan begin mathsize 14px style g colon R rightwards arrow R end style dengan begin mathsize 14px style f open parentheses x close parentheses equals square root of 2 x end root end style dan begin mathsize 14px style g open parentheses x close parentheses equals x squared minus 1 end style. NIlai dari begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses square root of 3 over 2 end root close parentheses end style adalah....

  1. begin mathsize 14px style negative 2 end style

  2. begin mathsize 14px style 2 end style

  3. begin mathsize 14px style plus-or-minus 2 end style

  4. begin mathsize 14px style 3 end style

  5. begin mathsize 14px style 4 end style

Pembahasan Soal:

Pertama kita menetukan fungsi komposisi:

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 cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses x squared minus 1 close parentheses end cell row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell square root of 2 open parentheses x squared minus 1 close parentheses end root end cell end table end style

kemudian kita menetukan invers dari fungsi komposisi tersebut:

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 square root of 2 open parentheses x squared minus 1 close parentheses end root end cell row y equals cell square root of 2 open parentheses x squared minus 1 close parentheses end root end cell row cell y squared end cell equals cell 2 open parentheses x squared minus 1 close parentheses end cell row cell y squared end cell equals cell 2 x squared minus 2 end cell row cell 2 x squared end cell equals cell y squared plus 2 end cell row x equals cell square root of fraction numerator y squared plus 2 over denominator 2 end fraction end root 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 square root of fraction numerator x squared plus 2 over denominator 2 end fraction end root end cell end table end style

Sehingga nilai dari Error converting from MathML to accessible text. adalah:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes 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 square root of fraction numerator x squared plus 2 over denominator 2 end fraction end root end cell row cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent open parentheses square root of 3 over 2 end root close parentheses end cell equals cell square root of fraction numerator square root of begin display style 3 over 2 end style end root squared plus 2 over denominator 2 end fraction end root end cell row cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent open parentheses square root of 3 over 2 end root close parentheses end cell equals cell square root of fraction numerator begin display style 3 over 2 end style plus 2 over denominator 2 end fraction end root end cell row cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent open parentheses square root of 3 over 2 end root close parentheses end cell equals cell square root of 7 over 4 end root end cell row blank blank blank end table end style

Jadi, nilai undefined adalah begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 7 over 4 end root end cell end table end style

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Nasrullah

Mahasiswa/Alumni Universitas Negeri Makassar

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui fungsi f:R→R dan g:R→R. Jika fungsi  f(x)=3x−52x+7​,x=35​ dan g−1(x)=4x−1, Nilai fungsi  (f∘g)−1(2)adalah ...

Pembahasan Soal:

Pertama akan dicari invers dari fungsi  begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 2 x plus 7 over denominator 3 x minus 5 end fraction comma space x not equal to 5 over 3 end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator 2 x plus 7 over denominator 3 x minus 5 end fraction end cell row cell 3 y x minus 5 y end cell equals cell 2 x plus 7 end cell row cell 3 y x minus 2 x end cell equals cell 5 y plus 7 end cell row cell x open parentheses 3 y minus 2 close parentheses end cell equals cell 5 y plus 7 end cell row x equals cell fraction numerator 5 y plus 7 over denominator 3 y minus 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 5 x plus 7 over denominator 3 x minus 2 end fraction end cell end table end style 

Dengan menggunakan sifat invers akan ditentukan  begin mathsize 14px style left parenthesis f ring operator g right parenthesis 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 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 cell space space space space space space space space space space space space space space space space end cell equals cell g to the power of negative 1 end exponent open parentheses fraction numerator 5 x plus 7 over denominator 3 x minus 2 end fraction close parentheses end cell row cell space space space space space space space space space space space space space space space space end cell equals cell 4 open parentheses fraction numerator 5 x plus 7 over denominator 3 x minus 2 end fraction close parentheses minus 1 end cell row cell space space space space space space space space space space space space space space space space end cell equals cell fraction numerator 20 x plus 28 minus 3 x plus 2 over denominator 3 x minus 2 end fraction end cell row cell space space space space space space space space space space space space space space space space end cell equals cell fraction numerator 17 x plus 30 over denominator 3 x minus 2 end fraction end cell end table end style        

Nilai fugsi begin mathsize 14px style left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent open parentheses 2 close parentheses end style adalah  

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent open parentheses 2 close parentheses end cell equals cell fraction numerator 17 open parentheses 2 close parentheses plus 30 over denominator 3 open parentheses 2 close parentheses minus 2 end fraction end cell row cell space space space space space space space space space space space space space space space space space end cell equals cell fraction numerator 34 plus 30 over denominator 6 minus 2 end fraction end cell row cell space space space space space space space space space space space space space space space space space end cell equals cell 64 over 4 end cell row cell space space space space space space space space space space space space space space space space space end cell equals 16 end table end style  

Jadi, jawaban yang tepat  adalah E

1

Roboguru

Diketahui fungsi f:R→R dan g:R→R. Jika fungsi f(x)=3x−52x+7​,x=35​ dan g−1(x)=4x−1, nilai fungsi (f∘g)−1(2) adalah...

Pembahasan Soal:

Dengan menggunakan konsep invers fungsi,

Misalkan begin mathsize 14px style f open parentheses x close parentheses equals y end style maka

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals y row cell fraction numerator 2 x plus 7 over denominator 3 x minus 5 end fraction end cell equals y row cell 2 x plus 7 end cell equals cell y open parentheses 3 x minus 5 close parentheses end cell row cell 2 x plus 7 end cell equals cell 3 x y minus 5 y end cell row cell 2 x minus 3 x y end cell equals cell negative 5 y minus 7 end cell row cell x open parentheses 2 minus 3 y close parentheses end cell equals cell negative 5 y minus 7 end cell row x equals cell fraction numerator negative 5 y minus 7 over denominator 2 minus 3 y end fraction comma y not equal to 2 over 3 end cell row x equals cell fraction numerator 5 y plus 7 over denominator 3 y minus 2 end fraction comma y not equal to 2 over 3 end cell end table end style

Sehingga diperoleh begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 5 x plus 7 over denominator 3 x minus 2 end fraction comma x not equal to 2 over 3 end style.

Selanjutnya dengan menggunakan sifat invers komposisi fungsi maka

begin mathsize 14px style 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 g to the power of negative 1 end exponent open parentheses x close parentheses ring operator f to the power of negative 1 end exponent open parentheses x close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator 5 x plus 7 over denominator 3 x minus 2 end fraction close parentheses end cell row blank equals cell 4 open parentheses fraction numerator 5 x plus 7 over denominator 3 x minus 2 end fraction close parentheses minus 1 end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses end cell equals cell 4 open parentheses fraction numerator 5 open parentheses 2 close parentheses plus 7 over denominator 3 open parentheses 2 close parentheses minus 2 end fraction close parentheses minus 1 end cell row blank equals cell 4 open parentheses 17 over 4 close parentheses minus 1 end cell row blank equals cell 17 minus 1 end cell row blank equals 16 end table end style

Dengan demikian nilai dari begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses end style adalah begin mathsize 14px style 16 end style.

Oleh karena itu, jawaban yang benar adalah E.

 

0

Roboguru

2. Fungsi f:R→Rdang:R→R didefinisikan oleh f(x)=2x+11 dan g(x)=1−x. Tentukan: h. (g−1∘f−1)(x)

Pembahasan Soal:

Perhatikan penghitungan berikut!

 f open parentheses x close parentheses equals 2 x plus 11 misal colon space y equals f open parentheses x close parentheses space maka space x equals f to the power of negative 1 end exponent open parentheses x close parentheses f open parentheses x close parentheses equals 2 x plus 11 y equals 2 x plus 11 y minus 11 equals 2 x fraction numerator y minus 11 over denominator 2 end fraction equals x f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus 11 over denominator 2 end fraction  g open parentheses x close parentheses equals 1 minus x misal colon space z equals g open parentheses x close parentheses space maka space x equals g to the power of negative 1 end exponent open parentheses x close parentheses g open parentheses x close parentheses equals 1 minus x z equals 1 minus x x equals 1 minus z g to the power of negative 1 end exponent open parentheses x close parentheses equals 1 minus x  

sehingga, 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 yaitu:

 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 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 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 g to the power of negative 1 end exponent open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses 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 1 minus open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses 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 2 over 2 minus open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses 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 fraction numerator 2 minus x plus 11 over denominator 2 end fraction 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 fraction numerator 13 minus x over denominator 2 end fraction                  

Jadi, 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 fraction numerator 13 minus x over denominator 2 end fraction .space 

0

Roboguru

Diketahui f(x)=x−2 dan g(x)=3x+1, maka (f∘g)−1(x)=...

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

Jika f(x)=5x−7 dan g(x)=8−6x, tentukan b. Invers fungsi dari g komposisi .

Pembahasan Soal:

Diketahui:

begin mathsize 14px style f open parentheses x close parentheses equals 5 x minus 7 end style

begin mathsize 14px style g open parentheses x close parentheses equals 8 minus 6 x end style

Akan dicari invers fungsi dari size 14px g komposisi size 14px g, tentukan terlebih dahulu begin mathsize 14px style open parentheses g ring operator g close parentheses open parentheses x close parentheses end style sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses g ring operator g close parentheses open parentheses x close parentheses end cell equals cell g open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell g open parentheses 8 minus 6 x close parentheses end cell row blank equals cell 8 minus 6 open parentheses 8 minus 6 x close parentheses end cell row blank equals cell 8 minus 48 plus 36 x end cell row blank equals cell 36 x minus 40 end cell end table end style 

Kemudian tentukan invers fungsi komposisi tersebut, diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses g ring operator g close parentheses open parentheses x close parentheses end cell equals cell 36 x minus 40 space space space open parentheses ubah space open parentheses g ring operator g close parentheses open parentheses x close parentheses space menjadi space y close parentheses end cell row y equals cell 36 x minus 40 end cell row cell y plus 40 end cell equals cell 36 x end cell row cell 36 x end cell equals cell y plus 40 end cell row x equals cell fraction numerator y plus 40 over denominator 36 end fraction space space space open parentheses ubah space x space menjadi space open parentheses g ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row cell open parentheses g 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 40 over denominator 36 end fraction end cell end table end style 

Jadi, invers fungsinya adalah begin mathsize 14px style space open parentheses g ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x plus 40 over denominator 36 end fraction end style.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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