Roboguru

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Pertanyaan

Diketahui : 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 fraction numerator x over denominator x 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 2 x end cell end table 

Tentukan : g left parenthesis x right parenthesis equals... ?

Catatan : 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  

Pembahasan Soal:

Perhatikan bahwa

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 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 cell fraction numerator x over denominator x minus 2 end fraction end cell equals cell g to the power of negative 1 end exponent open parentheses 2 x close parentheses end cell end table

Misalkan y equals 2 x space left right arrow y over 2 equals x. Maka dapat diperoleh

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

Selanjutnya dimisalkan z equals g to the power of negative 1 end exponent open parentheses x close parentheses, maka didapat

table attributes columnalign right center left columnspacing 0px end attributes row z equals cell g to the power of negative 1 end exponent open parentheses x close parentheses end cell row z equals cell fraction numerator x over denominator x minus 4 end fraction end cell row cell z open parentheses x minus 4 close parentheses end cell equals x row cell z x minus 4 z end cell equals x row cell z x minus x end cell equals cell 4 z end cell row cell x left parenthesis z minus 1 right parenthesis end cell equals cell 4 z end cell row x equals cell fraction numerator 4 z over denominator z minus 1 end fraction end cell row cell g open parentheses z close parentheses end cell equals cell fraction numerator 4 z over denominator z minus 1 end fraction space left right arrow space g open parentheses x close parentheses equals fraction numerator 4 x over denominator x minus 1 end fraction end cell end table 

Jadi, diperoleh g open parentheses x close parentheses equals fraction numerator 4 x over denominator x minus 1 end fraction 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

K. Prameswari

Mahasiswa/Alumni Universitas Gadjah Mada

Terakhir diupdate 03 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui  dan , tentukan : a.  dan   b.

Pembahasan Soal:

Bentuk umum invers dari fungsi rasional f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction adalah 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.

Diketahui fungsi f left parenthesis x right parenthesis equals fraction numerator 5 x minus 3 over denominator x plus 2 end fraction comma space x not equal to negative 2. Fungsi inversnya adalah f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction.

Bentuk umum invers dari fungsi linear g left parenthesis x right parenthesis equals a x plus b adalah g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x minus b over denominator a end fraction.

Diketahui pula fungsi g left parenthesis x right parenthesis equals 6 x minus 2. Fungsi inversnya adalah :

table attributes columnalign right center left columnspacing 0px end attributes row cell g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator x minus left parenthesis negative 2 right parenthesis over denominator 6 end fraction end cell row blank equals cell fraction numerator x plus 2 over denominator 6 end fraction end cell row blank equals cell 1 over 6 x plus 2 over 6 end cell end table 

Invers fungsi komposisi dari kedua fungsi di atas dapat dihitung seperti berikut,

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 left parenthesis x right parenthesis 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 left parenthesis x right parenthesis end cell row blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator 2 x minus 3 over denominator x minus 5 end fraction close parentheses end cell row blank equals cell 1 over 6 open parentheses fraction numerator 2 x minus 3 over denominator x minus 5 end fraction close parentheses plus 2 over 6 end cell row blank equals cell fraction numerator 2 x minus 3 over denominator 6 left parenthesis x minus 5 right parenthesis end fraction plus 2 over 6 end cell row blank equals cell fraction numerator 2 x minus 3 plus 2 left parenthesis x minus 5 right parenthesis over denominator 6 left parenthesis x minus 5 right parenthesis end fraction end cell row blank equals cell fraction numerator 2 x minus 3 plus 2 x minus 10 over denominator 6 x minus 30 end fraction end cell row blank equals cell fraction numerator 4 x minus 13 over denominator 6 x minus 30 end fraction end cell end table 

Jadi, fungsi invers f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative 2 x minus 3 over denominator x minus 5 end fractiong to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x plus 2 over denominator 6 end fraction, dan open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 4 x minus 13 over denominator 6 x minus 30 end fraction.

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Roboguru

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.

0

Roboguru

Diketahui dan . Nilai dari  adalah ....

Pembahasan Soal:

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 9 minus x close parentheses end cell row blank equals cell 2 open parentheses 9 minus x close parentheses plus 3 end cell row blank equals cell 18 minus 2 x plus 3 end cell row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell 21 minus 2 x end cell end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell row y equals cell 21 minus 2 x end cell row cell 2 x end cell equals cell 21 minus y end cell row x equals cell fraction numerator 21 minus y 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 y close parentheses end cell equals cell fraction numerator 21 minus y 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 21 minus x 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 11 close parentheses end cell equals cell fraction numerator 21 minus 11 over denominator 2 end fraction end cell row blank equals cell 10 over 2 end cell row blank equals 5 end table end style 

Maka nilai dari begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style adalah 5.

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Roboguru

Diketahui  dan . Jika fungsi  adalah ....

Pembahasan Soal:

Kita cari nilai komposisi begin mathsize 14px style open parentheses straight f ring operator straight g close parentheses open parentheses straight x close parentheses end style terlebih dahulu 
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses straight f ring operator straight g close parentheses open parentheses straight x close parentheses end cell equals cell straight f open parentheses straight g open parentheses straight x close parentheses close parentheses end cell row blank equals cell 4 open parentheses straight g open parentheses straight x close parentheses close parentheses plus 7 end cell row blank equals cell 4 open parentheses 3 straight x minus 11 close parentheses plus 7 end cell row blank equals cell 12 straight x minus 44 plus 7 end cell row blank equals cell 12 straight x minus 37 end cell end table end style 
 

selanjutnya kita tentukan nilai invers komposisi begin mathsize 14px style open parentheses straight f ring operator straight g close parentheses to the power of negative 1 end exponent open parentheses straight x close parentheses end style
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row straight y equals cell open parentheses straight f ring operator straight g close parentheses open parentheses straight x close parentheses equals 12 straight x minus 37 end cell row straight y equals cell 12 straight x minus 37 end cell row cell negative 12 straight x end cell equals cell negative 37 minus straight y end cell row straight x equals cell fraction numerator negative 37 minus straight y over denominator negative 12 end fraction end cell row straight x equals cell fraction numerator 37 plus straight y over denominator 12 end fraction end cell row cell straight f to the power of negative 1 end exponent open parentheses straight y close parentheses end cell equals cell fraction numerator 37 plus straight y over denominator 12 end fraction end cell row cell straight f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 37 plus straight x over denominator 12 end fraction end cell end table end style  
 

jadi fungsi begin mathsize 14px style open parentheses straight f ring operator straight g close parentheses to the power of negative 1 end exponent equals open parentheses straight x close parentheses end style dari  begin mathsize 14px style straight f open parentheses straight x close parentheses equals 4 straight x plus 7 end style dan begin mathsize 14px style straight g open parentheses straight x close parentheses equals 3 straight x minus 11 end style adalah begin mathsize 14px style fraction numerator 37 plus x over denominator 12 end fraction end style 

0

Roboguru

Diketahui dan . Nilai dari  adalah...

Pembahasan Soal:

Ingat kembali sifat invers fungsi komposisi:

space left parenthesis f ring operator g right parenthesis 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 space

Maka yang perlu kita tentukan terlebih dahulu adalah begin mathsize 14px style space 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 cell space 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 cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell open parentheses g left parenthesis x right parenthesis close parentheses minus 5 end cell row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell open parentheses 2 x plus 6 close parentheses minus 5 end cell row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell 2 x plus 1 end cell end table end style

Kemudian kita menetukan undefined

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 1 end cell row y equals cell 2 x plus 1 end cell row cell 2 x end cell equals cell y minus 1 end cell row x equals cell fraction numerator y minus 1 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 1 over denominator 2 end fraction end cell end table 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 to the power of negative 1 end exponent left parenthesis 5 right parenthesis end cell equals cell fraction numerator 5 minus 1 over denominator 2 end fraction end cell row blank equals cell 4 over 2 end cell row blank equals 2 end table end style

Jadi, nilai 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 5 right parenthesis end cell end table end style adalah 2

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Roboguru

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