Pertanyaan

Tentukan jika diketahui f ( x ) = x + 1 3 dan g ( x ) = x − 1 1 !

Tentukan left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis space dan space left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis jika diketahui $f (x) = \frac{3}{x + 1} dan g (x) = \frac{1}{x - 1}$ !

H. Hermawan

Master Teacher

Mahasiswa/Alumni Universitas Lampung

Jawaban terverifikasi

Pembahasan

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

Dengan menggunakan konsep invers fungsi komposisi:

$table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell row blank equals cell fraction numerator 3 over denominator begin display style fraction numerator 1 over denominator x minus 1 end fraction end style plus 1 end fraction end cell row blank equals cell fraction numerator 3 over denominator begin display style fraction numerator 1 plus x minus 1 over denominator x minus 1 end fraction end style end fraction end cell row blank equals cell fraction numerator 3 left parenthesis x minus 1 right parenthesis over denominator x end fraction end cell end table$

Sehingga,

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

Selanjutnya,

$table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals cell g left parenthesis f left parenthesis x right parenthesis right parenthesis end cell row blank equals cell fraction numerator 1 over denominator fraction numerator 3 over denominator x plus 1 end fraction minus 1 end fraction end cell row blank equals cell fraction numerator 1 over denominator fraction numerator 3 minus x minus 1 over denominator x plus 1 end fraction end fraction end cell row blank equals cell fraction numerator x plus 1 over denominator 2 minus x end fraction end cell end table$

Sehingga,

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

Dengan demikian,

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

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 1 minus 2 x over denominator negative x minus 1 end fraction end cell end table$