Roboguru

Diketahui fungsi f dan g adalah fungsi bijektif yang ditentukan dengan f(x)=2x+5 dan g(x)=x−2. Tentukan (g−1∘f−1)(x) dan (f−1∘g−1)(x).

Pertanyaan

Diketahui fungsi begin mathsize 14px style f end style dan size 14px g adalah fungsi bijektif yang ditentukan dengan begin mathsize 14px style f left parenthesis x right parenthesis equals 2 x plus 5 end style dan begin mathsize 14px style g left parenthesis x right parenthesis equals x minus 2 end style. Tentukan begin mathsize 14px style 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 style dan begin mathsize 14px style 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 style.space 

Pembahasan:

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

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

b. begin mathsize 14px style left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end style artinya size 14px f dimasukkan ke begin mathsize 14px style g end style 

Ingat!

begin mathsize 14px style 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 equals open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses 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 equals open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style 

Tentukan begin mathsize 14px style open parentheses f ring operator g close parentheses left parenthesis x right parenthesis end style dan begin mathsize 14px style left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end style terlebih dahulu.

begin mathsize 14px style 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 minus 2 close parentheses end cell row blank equals cell 2 open parentheses x minus 2 close parentheses plus 5 end cell row blank equals cell 2 x minus 4 plus 5 end cell row blank equals cell 2 x plus 1 end cell row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell g open parentheses 2 x plus 5 close parentheses end cell row blank equals cell open parentheses 2 x plus 5 close parentheses minus 2 end cell row blank equals cell 2 x plus 5 minus 2 end cell row blank equals cell 2 x plus 3 end cell end table end style 

Misalkan begin mathsize 14px style left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals m end style, maka begin mathsize 14px style 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 style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row 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 equals cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses comma space maka colon end cell row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell 2 x plus 1 end cell row m equals cell 2 x plus 1 end cell row cell m minus 1 end cell equals cell 2 x end cell row cell fraction numerator m minus 1 over denominator 2 end fraction end cell equals x row x equals cell fraction numerator m minus 1 over denominator 2 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 x minus 1 over denominator 2 end fraction end cell row 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 equals cell fraction numerator x minus 1 over denominator 2 end fraction end cell end table end style  

Misalkan begin mathsize 14px style left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis equals n end style, maka begin mathsize 14px style 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 style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row 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 equals cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell 2 x plus 3 end cell row n equals cell 2 x plus 3 end cell row cell n minus 3 end cell equals cell 2 x end cell row cell fraction numerator n minus 3 over denominator 2 end fraction end cell equals x row x equals cell fraction numerator n minus 3 over denominator 2 end fraction end cell 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 fraction numerator x minus 3 over denominator 2 end fraction end cell row 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 equals cell fraction numerator x minus 3 over denominator 2 end fraction end cell end table end style 

Jadi, begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank 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 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x minus 1 over denominator 2 end fraction end cell end table end style dan begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank 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 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x minus 3 over denominator 2 end fraction end cell end table end style.

Jawaban terverifikasi

Dijawab oleh:

A. Acfreelance

Mahasiswa/Alumni Universitas Indraprasta PGRI

Terakhir diupdate 06 Oktober 2021

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