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Tentukan invers dari fungsi :

Pertanyaan

Tentukan invers dari fungsi :

f left parenthesis x right parenthesis equals fraction numerator 3 x minus 2 over denominator 2 x plus 5 end fraction 

Pembahasan Soal:

Misalkan y equals f left parenthesis x right parenthesis. Maka dapat diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell f left parenthesis x right parenthesis end cell row y equals cell fraction numerator 3 x minus 2 over denominator 2 x plus 5 end fraction end cell row cell y cross times open parentheses 2 x plus 5 close parentheses end cell equals cell fraction numerator 3 x minus 2 over denominator 2 x plus 5 end fraction cross times open parentheses 2 x plus 5 close parentheses end cell row cell 2 x y plus 5 y end cell equals cell 3 x minus 2 end cell row cell 2 x y minus 3 x end cell equals cell negative 2 minus 5 y end cell row cell x open parentheses 2 y minus 3 close parentheses end cell equals cell negative 2 minus 5 y end cell row x equals cell fraction numerator negative 2 minus 5 y over denominator 2 y minus 3 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell fraction numerator negative 2 minus 5 y over denominator 2 y minus 3 end fraction space left right arrow space f to the power of negative 1 end exponent left parenthesis x right parenthesis equals equals fraction numerator negative 2 minus 5 x over denominator 2 x minus 3 end fraction end cell end table 

Jadi, invers fungsi di atas adalah f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative 2 minus 5 x over denominator 2 x minus 3 end fraction 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

K. Prameswari

Mahasiswa/Alumni Universitas Gadjah Mada

Terakhir diupdate 03 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Fungsi didefinisikan oleh . jika  invers dari, tentukan :

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell fraction numerator x plus 4 over denominator 2 x minus 3 end fraction comma space end cell row y equals cell fraction numerator x plus 4 over denominator 2 x minus 3 end fraction end cell row cell y left parenthesis 2 x minus 3 right parenthesis end cell equals cell x plus 4 end cell row cell 2 x y minus 3 y end cell equals cell x plus 4 end cell row cell 2 x y minus x end cell equals cell 4 plus 3 y end cell row cell x left parenthesis 2 y minus 1 right parenthesis end cell equals cell 4 plus 3 y end cell row x equals cell fraction numerator 4 plus 3 y over denominator 2 y minus 1 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 4 plus 3 x over denominator 2 x minus 1 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 3 x plus 4 over denominator 2 x minus 1 end fraction end cell end table

Jadi, Invers dari f left parenthesis x right parenthesis equals fraction numerator x plus 4 over denominator 2 x minus 3 end fraction comma space x not equal to negative 1 half adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 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 3 x plus 4 over denominator 2 x minus 1 end fraction end cell end table.

0

Roboguru

Jika , fungsi  =....

Pembahasan Soal:

Berdasarkan sifat fungsi invers, maka didapatkan :

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell fraction numerator x plus 3 over denominator x minus 5 end fraction end cell row y equals cell fraction numerator x plus 3 over denominator x minus 5 end fraction end cell row cell y left parenthesis x minus 5 right parenthesis end cell equals cell x plus 3 end cell row cell y x minus 5 y end cell equals cell x plus 3 end cell row cell y x minus x end cell equals cell 5 y plus 3 end cell row cell x left parenthesis y minus 1 right parenthesis end cell equals cell 5 y plus 3 end cell row x equals cell fraction numerator 5 y plus 3 over denominator y minus 1 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x plus 3 over denominator x minus 1 end fraction end cell end table

Dengan demikian, fungsi f to the power of negative 1 end exponent left parenthesis x right parenthesis adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 5 x plus 3 over denominator x minus 1 end fraction end cell end table.

Oleh karena itu, jawaban yang benar adalah C.

0

Roboguru

Diketahui fungsi f(x)=2x+1, g(x)=2x−1, maka nilai f−1(4) ....

Pembahasan Soal:

Dengan menggunakan konsep invers dari fungsi, maka:

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

Dengan demikian, hasil dari f1(4)=121.

Oleh karena itu, jawaban yang tepat adalah C.

0

Roboguru

2. Diketahui . Tentukan : b.

Pembahasan Soal:

Diketahui begin mathsize 14px style straight g left parenthesis straight x right parenthesis equals 3 straight x plus 5 end style maka inversnya

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row straight y equals cell 3 straight x plus 5 end cell row straight x equals cell 3 straight y plus 5 end cell row cell straight x minus 5 end cell equals cell 3 straight y end cell row cell fraction numerator straight x minus 5 over denominator 3 end fraction end cell equals straight y row cell straight g to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell fraction numerator straight x minus 5 over denominator 3 end fraction end cell end table end style

Dengan demikian, invers dari  begin mathsize 14px style straight g left parenthesis straight x right parenthesis equals 3 straight x plus 5 end style adalah g1(x)=3x5.

0

Roboguru

Diketahui  dan   b.  ?

Pembahasan Soal:

Dengan menggunakan rumus invers fungsi

begin mathsize 14px style z open parentheses x close parentheses equals a x plus b blank semicolon a not equal to 0 z to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus b over denominator a end fraction blank semicolon a not equal to 0 end style 

Diperoleh 

begin mathsize 14px style f open parentheses x close parentheses equals x minus 3 f to the power of negative 1 end exponent open parentheses x close parentheses equals x plus 3 g open parentheses x close parentheses equals 2 x plus 4 g to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus 4 over denominator 2 end fraction end style  

Nilai begin mathsize 14px style left parenthesis g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis end style adalah begin mathsize 14px style f to the power of negative 1 end exponent end style dimasukkan ke begin mathsize 14px style g to the power of negative 1 end exponent space colon space left parenthesis g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis equals g to the power of negative 1 end exponent left parenthesis f to the power of negative 1 end exponent left parenthesis x right parenthesis right parenthesis end style 

begin mathsize 14px style left parenthesis g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis equals fraction numerator left parenthesis x plus 3 right parenthesis minus 4 over denominator 2 end fraction left parenthesis g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis equals fraction numerator x minus 1 over denominator 2 end fraction end style  

Jadi, diperoleh bahwa begin mathsize 14px style left parenthesis g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis equals fraction numerator x minus 1 over denominator 2 end fraction end style                undefined 

0

Roboguru

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