Diketahui f(x)=x+35x−2​, dan g(x)=2x+7 Tentukan (f−1∘g−1)(x) dan (g−1∘f−1)(x)

Pertanyaan

Diketahui f open parentheses x close parentheses equals fraction numerator 5 x minus 2 over denominator x plus 3 end fraction, dan g open parentheses x close parentheses equals 2 x plus 7

Tentukan 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 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

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

 table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent left parenthesis x right parenthesis 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 negative 3 x plus 17 over denominator x minus 17 end fraction end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent left parenthesis x right parenthesis 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 negative 10 x plus 33 over denominator 2 x minus 10 end fraction end cell end table 

Pembahasan

Diketahui f open parentheses x close parentheses equals fraction numerator 5 x minus 2 over denominator x plus 3 end fraction, dan g open parentheses x close parentheses equals 2 x plus 7. Dalam fungsi, berlaku hubungan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis 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 f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell end table dan  table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis 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 g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell end table dan , maka:

  • table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell end table 

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses left parenthesis x right parenthesis end cell equals cell g left parenthesis f left parenthesis x right parenthesis right parenthesis end cell row blank equals cell 2 open parentheses fraction numerator 5 x minus 2 over denominator x plus 3 end fraction close parentheses plus 7 end cell row blank equals cell fraction numerator 10 x minus 4 over denominator x plus 3 end fraction plus 7 end cell row blank equals cell fraction numerator 10 x minus 4 plus 7 left parenthesis x plus 3 right parenthesis over denominator x plus 3 end fraction end cell row blank equals cell fraction numerator 10 x minus 4 plus 7 x plus 21 over denominator x plus 3 end fraction end cell row blank equals cell fraction numerator 17 x plus 17 over denominator x plus 3 end fraction end cell end table 

Untuk menentukan inversnya, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses left parenthesis x right parenthesis end cell equals cell fraction numerator 17 x plus 17 over denominator x plus 3 end fraction end cell row y equals cell fraction numerator 17 x plus 17 over denominator x plus 3 end fraction end cell row cell y left parenthesis x plus 3 right parenthesis end cell equals cell 17 x plus 17 end cell row cell x y plus 3 y end cell equals cell 17 x plus 17 end cell row cell x y minus 17 x end cell equals cell negative 3 y plus 17 end cell row cell x left parenthesis y minus 17 right parenthesis end cell equals cell negative 3 y plus 17 end cell row x equals cell fraction numerator negative 3 y plus 17 over denominator y minus 17 end fraction end cell row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator negative 3 x plus 17 over denominator x minus 17 end fraction end cell row cell f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator negative 3 x plus 17 over denominator x minus 17 end fraction end cell end table 

  • table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell end table 

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses left parenthesis x right parenthesis end cell equals cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell row blank equals cell fraction numerator 5 left parenthesis 2 x plus 7 right parenthesis minus 2 over denominator left parenthesis 2 x plus 7 right parenthesis plus 3 end fraction end cell row blank equals cell fraction numerator 10 x plus 35 minus 2 over denominator 2 x plus 7 plus 3 end fraction end cell row blank equals cell fraction numerator 10 x plus 33 over denominator 2 x plus 10 end fraction end cell end table 

Untuk menentukan inversnya, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses left parenthesis x right parenthesis end cell equals cell fraction numerator 10 x plus 33 over denominator 2 x plus 10 end fraction end cell row y equals cell fraction numerator 10 x plus 33 over denominator 2 x plus 10 end fraction end cell row cell y left parenthesis 2 x plus 10 right parenthesis end cell equals cell 10 x plus 33 end cell row cell 2 x y plus 10 y end cell equals cell 10 x plus 33 end cell row cell 2 x y minus 10 x end cell equals cell negative 10 y plus 33 end cell row cell x left parenthesis 2 y minus 10 right parenthesis end cell equals cell negative 10 y plus 33 end cell row x equals cell fraction numerator negative 10 y plus 33 over denominator 2 y minus 10 end fraction end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator negative 10 x plus 33 over denominator 2 x minus 10 end fraction end cell row cell g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator negative 10 x plus 33 over denominator 2 x minus 10 end fraction end cell end table 

Jadi, table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent left parenthesis x right parenthesis 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 negative 3 x plus 17 over denominator x minus 17 end fraction end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent left parenthesis x right parenthesis 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 negative 10 x plus 33 over denominator 2 x minus 10 end fraction end cell end table 

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