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Tentukan ( f ∘ g ) − 1 ( x ) dan ( g ∘ f ) − 1 ( x ) jika diketahui f ( x ) = x − 5 3 ​ dan g ( x ) = 6 − 2 x !

Tentukan  jika diketahui  !

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Z. Apriani

Master Teacher

Mahasiswa/Alumni Institut Teknologi Bandung

Jawaban terverifikasi

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Pembahasan

Dengan menggunakan konsepinvers fungsi komposisi: Sehingga, Selanjutnya, Sehingga, Jadi,

Dengan menggunakan konsep invers fungsi komposisi:

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 left parenthesis x right parenthesis right parenthesis end cell row blank equals cell fraction numerator 3 over denominator 6 minus 2 x minus 5 end fraction end cell row blank equals cell fraction numerator 3 over denominator 1 minus 2 x end fraction end cell end table

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 fraction numerator 3 over denominator 1 minus 2 x end fraction end cell row y equals cell fraction numerator 3 over denominator 1 minus 2 x end fraction end cell row cell y left parenthesis 1 minus 2 x right parenthesis end cell equals 3 row cell y minus 2 y x end cell equals 3 row cell negative 2 y x end cell equals cell 3 minus y end cell row x equals cell fraction numerator 3 minus y over denominator negative 2 y end fraction end cell row cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 3 minus x over denominator negative 2 x end fraction end cell end table

Selanjutnya,

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f right parenthesis 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 6 minus 2 open parentheses fraction numerator 3 over denominator x minus 5 end fraction close parentheses end cell row blank equals cell 6 minus fraction numerator 6 over denominator x minus 5 end fraction end cell row blank equals cell fraction numerator 6 left parenthesis x minus 5 right parenthesis minus 6 over denominator x minus 5 end fraction end cell row blank equals cell fraction numerator 6 x minus 30 minus 6 over denominator x minus 5 end fraction end cell row blank equals cell fraction numerator 6 x minus 36 over denominator x minus 5 end fraction end cell end table

Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator 6 x minus 36 over denominator x minus 5 end fraction end cell row cell y left parenthesis x minus 5 right parenthesis end cell equals cell 6 x minus 36 end cell row cell y x minus 6 x minus 5 y end cell equals cell negative 36 end cell row cell x left parenthesis y minus 6 right parenthesis end cell equals cell 5 y minus 36 end cell row x equals cell fraction numerator 5 y minus 36 over denominator y minus 6 end fraction end cell row cell left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x minus 36 over denominator x minus 6 end fraction end cell end table


Jadi, 

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis f ring operator g right parenthesis 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 3 minus x over denominator negative 2 x end fraction end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis g ring operator f right parenthesis 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 5 x minus 36 over denominator x minus 6 end fraction end cell end table

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