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

Pertanyaan

Diketahui fungsi begin mathsize 14px style f open parentheses x close parentheses equals 4 x plus 2 end style dan begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator x minus 3 over denominator x plus 1 end fraction end style, size 14px x size 14px not equal to size 14px 1. Nilai dari begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses equals... end stylebegin mathsize 14px style space end style 

  1. begin mathsize 14px style negative 1 third end style 

  2. size 14px 3 

  3. begin mathsize 14px style 4 end style 

  4. size 14px 5 

  5. begin mathsize 14px style 10 end style 

A. Armanda

Master Teacher

Mahasiswa/Alumni Universitas Indraprasta PGRI

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah B.

Pembahasan

Fungsi komposisi merupakan penggabungan operasi dua jenis fungsi begin mathsize 14px style f left parenthesis x right parenthesis end style dan size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis sehingga menghasilkan sebuah fungsi baru. Operasi fungsi komposisi biasa dilambangkan dengan "size 14px ring operator" dan dibaca komposisi atau bundaran. Fungsi baru yang dapat terbentuk dari begin mathsize 14px style f left parenthesis x right parenthesis end style dan size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis adalah:

a. begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style artinya size 14px g dimasukkan ke begin mathsize 14px style f end style 

b. begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses end style artinya begin mathsize 14px style f end style dimasukkan ke size 14px g 

Sehingga,

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 f left parenthesis g open parentheses x close parentheses right parenthesis end cell row blank equals cell f left parenthesis fraction numerator x minus 3 over denominator x plus 1 end fraction right parenthesis end cell row blank equals cell 4 left parenthesis fraction numerator x minus 3 over denominator x plus 1 end fraction right parenthesis plus 2 end cell row blank equals cell fraction numerator 4 x minus 12 over denominator x plus 1 end fraction plus 2 end cell row blank equals cell fraction numerator 4 x minus 12 plus 2 left parenthesis x plus 1 right parenthesis over denominator x plus 1 end fraction end cell row blank equals cell fraction numerator 4 x minus 12 plus 2 x plus 2 over denominator x plus 1 end fraction end cell row blank equals cell fraction numerator 6 x minus 10 over denominator x plus 1 end fraction space end cell end table  

Misal, open parentheses f ring operator g close parentheses open parentheses x close parentheses equals y comma spacemaka invers fungsi komposisi tersebut:

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 6 x minus 10 over denominator x plus 1 end fraction end cell row y equals cell fraction numerator 6 x minus 10 over denominator x plus 1 end fraction end cell row cell y open parentheses x plus 1 close parentheses end cell equals cell 6 x minus 10 end cell row cell y x plus y end cell equals cell 6 x minus 10 end cell row cell y x minus 6 x end cell equals cell negative 10 minus y end cell row cell x open parentheses y minus 6 close parentheses end cell equals cell negative 10 minus y end cell row x equals cell fraction numerator negative 10 minus y over denominator y minus 6 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 negative 10 minus x over denominator x minus 6 end fraction space end cell end table  

Maka, nilai open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses spaceadalah:

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 negative 10 minus x over denominator x minus 6 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 negative 10 minus 2 over denominator 2 minus 6 end fraction end cell row blank equals cell fraction numerator negative 12 over denominator negative 4 end fraction end cell row blank equals cell 3 space end cell end table 

Jadi, jawaban yang tepat adalah B.

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