Jika f(x)=3x−4 dan g(x)=3−4x. Tentukanlah (g∘f)−1(x) dan (f∘g)−1(x).

Pertanyaan

Jika f open parentheses x close parentheses equals 3 x minus 4 dan g open parentheses x close parentheses equals 3 minus 4 x. Tentukanlah open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses dan open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses. 

H. Nur

Master Teacher

Mahasiswa/Alumni Universitas Jember

Jawaban terverifikasi

Jawaban

bentuk dari bold left parenthesis bold italic g bold ring operator bold italic f bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold italic x bold right parenthesis bold equalstable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold minus bold 13 over denominator bold 12 end fraction end cell end table dan 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 equalstable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold plus bold 5 over denominator bold 12 end fraction end cell end table.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah  bold left parenthesis bold italic g bold ring operator bold italic f bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold italic x bold right parenthesis bold equalstable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold minus bold 13 over denominator bold 12 end fraction end cell end table dan 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 equalstable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold plus bold 5 over denominator bold 12 end fraction 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 g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses dan 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:

f open parentheses x close parentheses equals 3 x minus 4 g open parentheses x close parentheses equals 3 minus 4 x

 Ditanya:

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

Mencari open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses dengan sifat invers open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses f to the power of negative 1 end exponent ring operator g 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 g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses fraction numerator 3 minus x over denominator 4 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator 3 minus x over denominator 4 end fraction end style close parentheses plus 4 over denominator 3 end fraction end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator 3 minus x over denominator 4 end fraction end style close parentheses minus begin display style 16 over 4 end style over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 3 minus x minus 16 over denominator 4 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative x minus 13 over denominator 4 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator negative x minus 13 over denominator 4 end fraction times 1 third end cell row blank equals cell fraction numerator negative x minus 13 over denominator 12 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 plus 4 over denominator 3 end fraction close parentheses end cell row blank equals cell fraction numerator 3 minus open parentheses begin display style fraction numerator x plus 4 over denominator 3 end fraction end style close parentheses over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style 9 over 3 end style minus open parentheses begin display style fraction numerator x plus 4 over denominator 3 end fraction end style close parentheses over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 9 minus x minus 4 over denominator 3 end fraction end style over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative x plus 5 over denominator 3 end fraction end style over denominator 4 end fraction end cell row blank equals cell fraction numerator negative x plus 5 over denominator 3 end fraction times 1 fourth end cell row blank equals cell fraction numerator negative x plus 5 over denominator 12 end fraction end cell end table

Dengan demikian, bentuk dari bold left parenthesis bold italic g bold ring operator bold italic f bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold italic x bold right parenthesis bold equalstable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold minus bold 13 over denominator bold 12 end fraction end cell end table dan 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 equalstable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold plus bold 5 over denominator bold 12 end fraction end cell end table.

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Pertanyaan serupa

Misal f : R→R dan g : R→R dengan f(x)=3x+1 dan g(x)=2−5x. Tentukanlah (g∘f)−1(x).

26

4.3

Jawaban terverifikasi

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