Fungsi f:R→R dinyatakan dengan f(x)=2x+13x−1​, x=−21​. Jika f−1 invers fungsi f, bentuk f−1(x) adalah …

Pertanyaan

Fungsi begin mathsize 14px style f colon straight R rightwards arrow straight R end style dinyatakan dengan begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 3 x minus 1 over denominator 2 x plus 1 end fraction comma space x not equal to negative 1 half end style. Jika begin mathsize 14px style f to the power of negative 1 end exponent end style invers fungsi size 14px f, bentuk begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses end style adalah begin mathsize 14px style horizontal ellipsis end style 

  1. begin mathsize 14px style fraction numerator x minus 1 over denominator 3 minus 2 x end fraction comma space x not equal to 3 over 2 end style

  2. begin mathsize 14px style fraction numerator x plus 1 over denominator 2 x minus 3 end fraction comma space x not equal to 3 over 2 end style

  3. fraction numerator size 14px x size 14px plus size 14px 1 over denominator size 14px 3 size 14px minus size 14px 2 size 14px x end fraction size 14px comma size 14px space size 14px x size 14px not equal to size 14px 3 over size 14px 2

  4. fraction numerator size 14px x size 14px minus size 14px 1 over denominator size 14px 2 size 14px x size 14px minus size 14px 3 end fraction size 14px comma size 14px space size 14px x size 14px not equal to size 14px 3 over size 14px 2

  5. fraction numerator size 14px x size 14px minus size 14px 1 over denominator size 14px 2 size 14px x size 14px plus size 14px 3 end fraction size 14px comma size 14px space size 14px x size 14px not equal to size 14px minus size 14px 3 over size 14px 2

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

Pembahasan

Ada 3 langkah untuk menentukan fungsi invers, yaitu:

1. Ubahlah bentuk begin mathsize 14px style y equals f open parentheses x close parentheses end style menjadi bentuk size 14px x size 14px equals size 14px f begin mathsize 14px style left parenthesis y right parenthesis end style.

2. Tuliskan begin mathsize 14px style x end style sebagai begin mathsize 14px style f to the power of negative 1 end exponent open parentheses y close parentheses end style sehingga begin mathsize 14px style f to the power of negative 1 end exponent left parenthesis y right parenthesis equals f left parenthesis y right parenthesis end style.

3. Ubahlah variabel y dengan x sehingga diperoleh rumus fungsi inversbegin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses end style.

Oleh karena itu, misalkan begin mathsize 14px style y equals f open parentheses x close parentheses end style. Maka

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

Karena penyebut tidak boleh bernilai nol, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 3 minus 2 y end cell not equal to 0 row 3 not equal to cell 2 y end cell row cell 3 over 2 end cell not equal to y end table end style

Dengan demikian, begin mathsize 14px style f to the power of negative 1 end exponent open parentheses y close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator y minus 1 over denominator 3 minus 2 y end fraction end cell end table comma space y not equal to 3 over 2 end style dan mengakibatkan begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x minus 1 over denominator 3 minus 2 x end fraction end cell end table comma space x not equal to 3 over 2 end style.

Jadi, jawaban yang tepat adalah A.

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