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Jika fungsi f(x)=x−52x+3​,x=5 dan g(x) = 3x + 1 maka (gof)−1(x) = ....

Pertanyaan

Jika fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 2 x plus 3 over denominator x minus 5 end fraction comma space x not equal to 5 end style dan g(x) = 3x + 1 maka begin mathsize 14px style left parenthesis g o f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end style = ....

  1. begin mathsize 14px style fraction numerator 5 x plus 4 over denominator x plus 7 end fraction comma space x not equal to negative 7 end style

  2. begin mathsize 14px style fraction numerator 5 x plus 7 over denominator x minus 4 end fraction comma space x not equal to 4 end style

  3. begin mathsize 14px style fraction numerator 5 x plus 4 over denominator x minus 7 end fraction comma space x not equal to 7 end style

  4. begin mathsize 14px style fraction numerator 5 x minus 4 over denominator x minus 7 end fraction comma space x not equal to 7 end style

  5. begin mathsize 14px style fraction numerator 5 x minus 7 over denominator x minus 4 end fraction comma space x not equal to 4 end style

Pembahasan Soal:

begin mathsize 14px style straight i right parenthesis space Cari space straight g ring operator straight f  space space space 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 3 straight f open parentheses straight x close parentheses plus 1 equals 3. fraction numerator 2 straight x plus 3 over denominator straight x minus 5 end fraction plus 1 equals fraction numerator 6 straight x plus 9 over denominator straight x minus 5 end fraction plus fraction numerator straight x minus 5 over denominator straight x minus 5 end fraction  space space space equals fraction numerator 6 straight x plus 9 plus straight x minus 5 over denominator straight x minus 5 end fraction equals fraction numerator 7 straight x plus 4 over denominator straight x minus 5 end fraction    ii right parenthesis space Cari space invers space dengan space formula space berikut  space space space space straight h open parentheses straight x close parentheses equals fraction numerator ax plus straight b over denominator cx plus straight d end fraction rightwards arrow straight h to the power of negative 1 end exponent open parentheses straight x close parentheses equals fraction numerator negative dx plus straight b over denominator cx minus straight a end fraction    space space space space open parentheses straight g ring operator straight f close parentheses open parentheses straight x close parentheses equals fraction numerator 7 straight x plus 4 over denominator straight x minus 5 end fraction  space space space space Dengan space straight a equals 7 comma space straight b equals 4 comma space straight c equals 1 comma space straight d equals negative 5 comma space maka  space space space space 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 5 straight x plus 4 over denominator straight x minus 7 end fraction end style

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. Munawaroh

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 05 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui f(x) = 3x + 4 dan g(x) = 2x+14x−5​;x=−21​. Invers (f∘g)(x) = ...

Pembahasan Soal:

begin mathsize 12px 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  equals 3 open parentheses fraction numerator 4 x minus 5 over denominator 2 x plus 1 end fraction close parentheses plus 4  equals open parentheses fraction numerator 12 x minus 15 over denominator 2 x plus 1 end fraction close parentheses plus 4 open parentheses fraction numerator 2 x plus 1 over denominator 2 x plus 1 end fraction close parentheses  equals open parentheses fraction numerator 12 x minus 15 over denominator 2 x plus 1 end fraction close parentheses plus open parentheses fraction numerator 8 x plus 4 over denominator 2 x plus 1 end fraction close parentheses  equals fraction numerator 20 x minus 11 over denominator 2 x plus 1 end fraction end style

Jika begin mathsize 12px style f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction comma space x not equal to negative d over c end style maka begin mathsize 12px 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 negative a over c end style

sehingga diperoleh

begin mathsize 12px style open parentheses f ring operator g close parentheses open parentheses x close parentheses equals fraction numerator 20 x minus 11 over denominator 2 x plus 1 end fraction comma space x not equal to negative 1 half end style maka;

begin mathsize 12px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative x minus 11 over denominator 2 x minus 20 end fraction equals fraction numerator x plus 11 over denominator negative 2 x plus 20 end fraction comma space x not equal to 10 end style

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Roboguru

Diketahui fungsi f(x)=3+2x, g(x)=x+43x−1​,x=−4. Invers dari (f∘g)(x)= ...

Pembahasan Soal:

Tentukan terlebih dahulu komposisi fungsi open parentheses f ring operator g close parentheses open parentheses x close parentheses, yaitu:

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 fraction numerator 3 x minus 1 over denominator x plus 4 end fraction close parentheses end cell row blank equals cell 3 plus 2 open parentheses fraction numerator 3 x minus 1 over denominator x plus 4 end fraction close parentheses end cell row blank equals cell 3 plus fraction numerator 6 x minus 2 over denominator x plus 4 end fraction end cell row blank equals cell fraction numerator 3 open parentheses x plus 4 close parentheses over denominator x plus 4 end fraction plus fraction numerator 6 x minus 2 over denominator x plus 4 end fraction end cell row blank equals cell fraction numerator 3 x plus 12 over denominator x plus 4 end fraction plus fraction numerator 6 x minus 2 over denominator x plus 4 end fraction end cell row blank equals cell fraction numerator 9 x plus 10 over denominator x plus 4 end fraction end cell end table

Kemudian tentukan invers dari 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 fraction numerator 9 x plus 10 over denominator x plus 4 end fraction end cell row y equals cell fraction numerator 9 x plus 10 over denominator x plus 4 end fraction end cell row cell y open parentheses x plus 4 close parentheses end cell equals cell 9 x plus 10 end cell row cell x y plus 4 y end cell equals cell 9 x plus 10 end cell row cell x y minus 9 x end cell equals cell negative 4 y plus 10 end cell row cell x open parentheses y minus 9 close parentheses end cell equals cell negative 4 y plus 10 end cell row x equals cell fraction numerator negative 4 y plus 10 over denominator y minus 9 end fraction end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent end cell equals cell fraction numerator negative 4 x plus 10 over denominator x minus 9 end fraction end cell end table

Maka, invers dari open parentheses f ring operator g close parentheses open parentheses x close parentheses equals fraction numerator negative 4 x plus 10 over denominator x minus 9 end fraction.

0

Roboguru

Diketahuif(x)=x+25x−3​,x=−2dang(x)=6x−2, tentukan : b. (fog)−1(x)dang−1(x)

Pembahasan Soal:

begin mathsize 14px style table row cell open parentheses f o 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 equals cell f open parentheses 6 x minus 2 close parentheses end cell blank blank row blank blank blank equals cell fraction numerator 5 open parentheses 6 x minus 2 close parentheses minus 3 over denominator 6 x minus 2 plus 2 end fraction end cell blank blank row blank blank blank equals cell fraction numerator 30 x minus 13 over denominator 6 x end fraction end cell blank blank row cell open parentheses f o g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 13 over denominator 6 x minus 30 end fraction end cell blank blank blank blank row cell g open parentheses x close parentheses to the power of negative 1 end exponent end cell equals cell fraction numerator x plus 2 over denominator 6 end fraction end cell atau cell 1 over 6 open parentheses x plus 2 close parentheses end cell blank blank end table end style

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Roboguru

Diketahui f(x) = 4x + 2 dan g(x)=x+1x−3​,x=−1. Invers dari (g∘f)(x) adalah...

Pembahasan Soal:

Fungsi komposisi g(x) dan f(x) dinyatakan begin mathsize 12px style open parentheses g ring operator f close parentheses left parenthesis x right parenthesis equals g open parentheses f open parentheses x close parentheses close parentheses end style

Langkah-langkah menentukan fungsi invers begin mathsize 12px style f to the power of negative 1 end exponent left parenthesis x right parenthesis end style jika f(x) telah diketahui:

  • mengubah persamaan y=f(x) dalam bentuk x sebagai fungsi y,
  • bentuk x sebagai fungsi y tersebut dinamakan begin mathsize 12px style f to the power of negative 1 end exponent left parenthesis y right parenthesis end style, dan
  • mengganti y pada begin mathsize 12px style f to the power of negative 1 end exponent left parenthesis y right parenthesis end style dengan x sehingga diperoleh begin mathsize 12px style f to the power of negative 1 end exponent left parenthesis x right parenthesis end style.

Diketahui:

f(x) = 4x + 2

begin mathsize 12px style g left parenthesis x right parenthesis equals fraction numerator x minus 3 over denominator x plus 1 end fraction comma space x not equal to negative 1 end style

Maka:

begin mathsize 12px style open parentheses g ring operator f close parentheses open parentheses x close parentheses equals g open parentheses f open parentheses x close parentheses close parentheses  equals g open parentheses 4 x plus 2 close parentheses  equals fraction numerator 4 x plus 2 minus 3 over denominator 4 x plus 2 plus 1 end fraction  equals fraction numerator 4 x minus 1 over denominator 4 x plus 3 end fraction end style

Invers dari begin mathsize 12px style open parentheses g ring operator f close parentheses left parenthesis x right parenthesis end style dapat ditemukan sebagai berikut:

begin mathsize 12px style y equals fraction numerator 4 x minus 1 over denominator 4 x plus 3 end fraction  y open parentheses 4 x plus 3 close parentheses equals 4 x minus 1  4 x y plus 3 y equals 4 x minus 1  4 x y minus 4 x equals negative 3 y minus 1  x open parentheses 4 y minus 4 close parentheses equals negative open parentheses 3 y plus 1 close parentheses  x equals fraction numerator negative open parentheses 3 y plus 1 close parentheses over denominator 4 y minus 4 end fraction  x equals fraction numerator negative open parentheses 3 y plus 1 close parentheses over denominator negative open parentheses 4 minus 4 y close parentheses end fraction  x equals fraction numerator 3 y plus 1 over denominator 4 minus 4 y end fraction  f to the power of negative 1 end exponent open parentheses y close parentheses equals fraction numerator 3 y plus 1 over denominator 4 minus 4 y end fraction end style

Jadi, begin mathsize 12px style open parentheses g ring operator f close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 3 x plus 1 over denominator 4 minus 4 x end fraction comma space x not equal to 1 end style

 

Tip!

Jika begin mathsize 12px style open parentheses g ring operator f close parentheses left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction end style maka begin mathsize 12px style open parentheses g ring operator f close parentheses 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 end style

begin mathsize 12px style open parentheses g ring operator f close parentheses left parenthesis x right parenthesis equals fraction numerator 4 x minus 1 over denominator 4 x plus 3 end fraction  a equals c equals 4  b equals negative 1  d equals 3  open parentheses g ring operator f close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative 3 x minus 1 over denominator 4 x minus 4 end fraction  equals fraction numerator negative open parentheses 3 x plus 1 close parentheses over denominator negative open parentheses 4 minus 4 x close parentheses end fraction  equals fraction numerator 3 x plus 1 over denominator 4 minus 4 x end fraction end style

0

Roboguru

Diketahui f(x)=x−14x−2​ dan g(x)=x+1. Nilai dari (f∘g)−1(2) adalah ...

Pembahasan Soal:

Tentukan terlebih dahulu fungsi komposisi 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 x plus 1 close parentheses end cell row blank equals cell fraction numerator 4 open parentheses x plus 1 close parentheses minus 2 over denominator x plus 1 minus 1 end fraction end cell row blank equals cell fraction numerator 4 x plus 4 minus 2 over denominator x end fraction end cell row blank equals cell fraction numerator 4 x plus 2 over denominator x end fraction end cell end table

Kemudian cari fungsi invers dari 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 fraction numerator 4 x plus 2 over denominator x end fraction end cell row y equals cell fraction numerator 4 x plus 2 over denominator x end fraction end cell row cell x y end cell equals cell 4 x plus 2 end cell row cell x y minus 4 x end cell equals 2 row cell x open parentheses y minus 4 close parentheses end cell equals 2 row x equals cell fraction numerator 2 over denominator y minus 4 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 2 over denominator x minus 4 end fraction end cell end table

Kemudian substitusi variabel x dengan 2 pada fungsi invers.

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 fraction numerator 2 over denominator x minus 4 end fraction 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 fraction numerator 2 over denominator 2 minus 4 end fraction end cell row blank equals cell fraction numerator 2 over denominator negative 2 end fraction end cell row blank equals cell negative 1 end cell end table

Maka, nilai dari open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses adalah negative 1.

0

Roboguru

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