Roboguru

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

Pertanyaan

Diketahui fungsi f open parentheses x close parentheses equals 3 plus 2 xg open parentheses x close parentheses equals fraction numerator 3 x minus 1 over denominator x plus 4 end fraction comma space x not equal to negative 4. Invers dari open parentheses f ring operator g close parentheses open parentheses x close parentheses equals ...

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.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Ayu

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

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

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) = 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

0

Roboguru

Diketahui fungsi f(x) = 3x + 4 dan g(x)=2x+14x−5​,x=−21​ invers (f o g) (x) adalah  ....

Pembahasan Soal:

begin mathsize 14px style open parentheses straight f ring operator straight g close parentheses open parentheses straight x close parentheses equals straight f open parentheses straight g open parentheses straight x close parentheses close parentheses  straight f open parentheses fraction numerator 4 straight x minus 5 over denominator 2 straight x plus 1 end fraction close parentheses equals 3 straight space open parentheses fraction numerator 4 straight x minus 5 over denominator 2 straight x plus 1 end fraction close parentheses plus 4  straight f open parentheses fraction numerator 4 straight x minus 5 over denominator 2 straight x plus 1 end fraction close parentheses equals fraction numerator 12 straight x minus 15 over denominator 2 straight x plus 1 end fraction plus 4  straight f open parentheses fraction numerator 4 straight x minus 5 over denominator 2 straight x plus 1 end fraction close parentheses equals fraction numerator 12 straight x minus 15 plus 4 left parenthesis 2 straight x plus 1 right parenthesis over denominator 2 straight x plus 1 end fraction  straight f open parentheses fraction numerator 4 straight x minus 5 over denominator 2 straight x plus 1 end fraction close parentheses equals fraction numerator 12 straight x minus 15 plus 8 straight x plus 4 over denominator 2 straight x plus 1 end fraction  straight f open parentheses fraction numerator 4 straight x minus 5 over denominator 2 straight x plus 1 end fraction close parentheses equals fraction numerator 20 straight x minus 11 over denominator 2 straight x plus 1 end fraction    Karena space straight f open parentheses fraction numerator 4 straight x minus 5 over denominator 2 straight x plus 1 end fraction close parentheses equals open parentheses straight f ring operator straight g close parentheses open parentheses straight x close parentheses space maka  open parentheses straight f ring operator straight g close parentheses open parentheses straight x close parentheses equals fraction numerator 20 straight x minus 11 over denominator 2 straight x plus 1 end fraction    Untuk space mencari space inversnya comma space misalkan space left parenthesis straight f ring operator straight g right parenthesis left parenthesis straight x right parenthesis equals straight y comma space maka  straight y equals fraction numerator 20 straight x minus 11 over denominator 2 straight x plus 1 end fraction rightwards double arrow straight x equals fraction numerator straight y plus 11 over denominator negative 2 straight y plus 20 end fraction  Maka comma  open parentheses straight f ring operator straight g close parentheses to the power of negative 1 end exponent open parentheses straight x close parentheses equals fraction numerator straight x plus 11 over denominator negative 2 straight x plus 20 end fraction end style

2

Roboguru

Fungsi f:R→Rdang:R→R dinyatakan oleh f(x)=x−3x+2​,x=3dang(x)=x+4., Invers fungsi (f∘g)(x)adalah(f∘g)−1(x)=....

Pembahasan Soal:

Ingat kembali cara cepat invers fungsi berikut:

f open parentheses x close parentheses equals fraction numerator a x plus b over denominator c x plus d end fraction rightwards double arrow f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative d x plus b over denominator c x minus a end fraction

Maka,

open parentheses f ring operator g close parentheses open parentheses x close parentheses equals fraction numerator x plus 6 over denominator x plus 1 end fraction rightwards double arrow 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 plus 6 over denominator x minus 1 end fraction comma space x not equal to 1

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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