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Diketahui dan . Fungsi invers dari adalah ....

Diketahui begin mathsize 14px style f left parenthesis x right parenthesis equals 2 x minus 1 end style dan begin mathsize 14px style g left parenthesis x right parenthesis equals fraction numerator x plus 3 over denominator 2 minus x end fraction end style. Fungsi invers dari begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style adalah ....

  1. begin mathsize 14px style fraction numerator 2 x plus 4 over denominator x plus 3 end fraction end style 

  2. fraction numerator size 14px 2 size 14px x size 14px minus size 14px 4 over denominator size 14px x size 14px plus size 14px 3 end fraction 

  3. fraction numerator size 14px 2 size 14px x size 14px plus size 14px 4 over denominator size 14px x size 14px minus size 14px 3 end fraction 

  4. fraction numerator size 14px minus size 14px 2 size 14px x size 14px plus size 14px 4 over denominator size 14px x size 14px plus size 14px 3 end fraction 

  5. begin mathsize 14px style fraction numerator negative 2 x minus 4 over denominator x minus 3 end fraction end style 

Jawaban:

Diketahui fungsi size 14px f size 14px left parenthesis size 14px x size 14px right parenthesis size 14px equals size 14px 2 size 14px x size 14px minus size 14px 1 dan begin mathsize 14px style g left parenthesis x right parenthesis equals fraction numerator x plus 3 over denominator 2 minus x end fraction end style, sehingga dapat ditentukan:

begin mathsize 14px style 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 open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell 2 open parentheses fraction numerator x plus 3 over denominator 2 minus x end fraction close parentheses minus 1 end cell row blank equals cell fraction numerator 2 x plus 6 over denominator 2 minus x end fraction minus 1 end cell row blank equals cell fraction numerator 2 x plus 6 over denominator 2 minus x end fraction minus fraction numerator 2 minus x over denominator 2 minus x end fraction end cell row blank equals cell fraction numerator 2 x plus 6 minus 2 plus x over denominator 2 minus x end fraction end cell row blank equals cell fraction numerator 3 x plus 4 over denominator 2 minus x end fraction end cell end table end style 

Selanjutnya, invers fungsi komposisi tersebut dapat ditentukan dengan menyatakan fungsi tersebut dalam variabel begin mathsize 14px style y end style seperti berikut.

begin mathsize 14px style 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 3 x plus 4 over denominator 2 minus x end fraction end cell row y equals cell fraction numerator 3 x plus 4 over denominator 2 minus x end fraction end cell row cell y open parentheses 2 minus x close parentheses end cell equals cell 3 x plus 4 end cell row cell 2 y minus x y end cell equals cell 3 x plus 4 end cell row cell 2 y minus 4 end cell equals cell 3 x plus x y end cell row cell 2 y minus 4 end cell equals cell x open parentheses 3 plus y close parentheses end cell row x equals cell fraction numerator 2 y minus 4 over denominator 3 plus y end fraction end cell end table end style 

Selanjutnya, dari persamaan yang diperoleh terakhir diubah dengan mengganti begin mathsize 14px style x space dengan space f to the power of negative 1 end exponent open parentheses x close parentheses end style dan begin mathsize 14px style y space dengan space x end style, sehingga diperoleh fungsi invers sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell fraction numerator 2 y minus 4 over denominator 3 plus y end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 2 x minus 4 over denominator 3 plus x end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 2 x minus 4 over denominator x plus 3 end fraction end cell end table end style  

Jadi, jawaban yang benar adalah B.

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