Misal f : R→R dan f : R→R dengan f(x)=4x+3 dan g(x)=x−1x​, x=1. Tentukanlah:  a. (f∘g)−1(x)  b.  (f∘g)−1(2)

Pertanyaan

Misal f space colon space R rightwards arrow R dan f space colon space R rightwards arrow R dengan f open parentheses x close parentheses equals 4 x plus 3 dan g open parentheses x close parentheses equals fraction numerator x over denominator x minus 1 end fraction comma space x not equal to 1.

Tentukanlah: 

a. open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses 

b.  open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses

H. Nur

Master Teacher

Mahasiswa/Alumni Universitas Jember

Jawaban terverifikasi

Jawaban

bentuk dari bold left parenthesis bold italic f bold ring operator bold italic g bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold italic x bold right parenthesis bold equalsfraction numerator bold italic x bold minus bold 3 over denominator bold italic x bold minus bold 7 end fraction dan table attributes columnalign right center left columnspacing 0px end attributes row cell bold left parenthesis bold italic f bold ring operator bold italic g bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold 2 bold right parenthesis end cell bold equals blank end tabletable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold 1 over bold 5 end cell end table.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah bold left parenthesis bold italic f bold ring operator bold italic g bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold italic x bold right parenthesis bold equalsfraction numerator bold italic x bold minus bold 3 over denominator bold italic x bold minus bold 7 end fraction dan table attributes columnalign right center left columnspacing 0px end attributes row cell bold left parenthesis bold italic f bold ring operator bold italic g bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold 2 bold right parenthesis end cell bold equals blank end tabletable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold 1 over bold 5 end cell end table.

Ingat!

Ada 2 cara menentukan formula invers fungsi komposisi, yaitu:

  1. Mula-mula menentukan fungsi komposisinya, kemudian diinverskan.
  2. Mula-mula menentukan invers masing-masing fungsi kemudian dikomposisikan.

Ingat pula open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals 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

Diketahui:

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell 4 x plus 3 end cell row cell g open parentheses x close parentheses end cell equals cell fraction numerator x over denominator x minus 1 end fraction comma space x not equal to 1 end cell end table

Ditanya:

a. open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses 

b.  open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses

Jawab:

Cari f 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 f open parentheses x close parentheses end cell equals cell 4 x plus 3 end cell row y equals cell 4 x plus 3 end cell row cell 4 x end cell equals cell y minus 3 end cell row x equals cell fraction numerator y minus 3 over denominator 4 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator y minus 3 over denominator 4 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 x minus 3 over denominator 4 end fraction end cell end table

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

Mencari open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses dengan sifat invers open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals 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:

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 blank 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 blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator x minus 3 over denominator 4 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator x minus 3 over denominator 4 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator x minus 3 over denominator 4 end fraction end style close parentheses minus 1 end fraction end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator x minus 3 over denominator 4 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator x minus 3 over denominator 4 end fraction end style close parentheses minus begin display style 4 over 4 end style end fraction end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator x minus 3 over denominator 4 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator x minus 3 minus 4 over denominator 4 end fraction end style close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator x minus 3 over denominator 4 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator x minus 7 over denominator 4 end fraction end style close parentheses end fraction end cell row blank equals cell fraction numerator x minus 3 over denominator up diagonal strike 4 end fraction times fraction numerator up diagonal strike 4 over denominator x minus 7 end fraction end cell row blank equals cell fraction numerator x minus 3 over denominator x minus 7 end fraction end cell end table

Mencari nilai open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses dengan substitusi x equals 2 pada 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 to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator x minus 3 over denominator x minus 7 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 minus 3 over denominator 2 minus 7 end fraction end cell row blank equals cell fraction numerator negative 1 over denominator negative 5 end fraction end cell row blank equals cell 1 fifth end cell end table

Dengan demikian, bentuk dari bold left parenthesis bold italic f bold ring operator bold italic g bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold italic x bold right parenthesis bold equalsfraction numerator bold italic x bold minus bold 3 over denominator bold italic x bold minus bold 7 end fraction dan table attributes columnalign right center left columnspacing 0px end attributes row cell bold left parenthesis bold italic f bold ring operator bold italic g bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold 2 bold right parenthesis end cell bold equals blank end tabletable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold 1 over bold 5 end cell end table.

36

5.0 (1 rating)

Pertanyaan serupa

Diketahui fungsi f(x)=x3 dan g(x)=3x−4. Jika a=(g−1∘f−1)(8), maka nilai (f−1∘g−1)(10a) adalah ...

34

5.0

Jawaban terverifikasi

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Roboguru

Coba GRATIS Aplikasi Ruangguru

Download di Google PlayDownload di AppstoreDownload di App Gallery

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ruangguru WhatsApp

081578200000

Email info@ruangguru.com

info@ruangguru.com

Contact 02140008000

02140008000

Ikuti Kami

©2022 Ruangguru. All Rights Reserved PT. Ruang Raya Indonesia