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Pertanyaan

Jika begin mathsize 14px style f left parenthesis x right parenthesis equals 4 to the power of 3 x minus 7 end exponent end style dan begin mathsize 14px style f to the power of negative 1 end exponent left parenthesis a right parenthesis equals 2 end style, maka nilai begin mathsize 14px style a end style yang memenuhi adalah ....

Pembahasan Soal:

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 4 to the power of 3 x minus 7 end exponent end cell row cell f to the power of negative 1 end exponent left parenthesis f left parenthesis x right parenthesis right parenthesis end cell equals cell x equals 2 end cell row cell f left parenthesis 2 right parenthesis end cell equals a row cell 4 to the power of 3 left parenthesis 2 right parenthesis minus 7 end exponent end cell equals a row a equals cell 1 fourth end cell end table end style  

Jadi, nilai begin mathsize 14px style a end style yang memenuhi adalah begin mathsize 14px style 1 fourth. end style 

Pembahasan terverifikasi oleh Roboguru

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Terakhir diupdate 30 Maret 2021

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Jika , maka

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x cubed plus 8 end cell row cell x cubed end cell equals cell y minus 8 end cell row x equals cell cube root of y minus 8 end root end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell cube root of x minus 8 end root end cell end table end style  

0

Roboguru

Diketahui  dan . Jika invers  adalah , maka  adalah ...

Pembahasan Soal:

begin mathsize 14px style f open parentheses x close parentheses equals 3 x plus 1 end style dan begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 3 end fraction comma space x not equal to negative 3 end style, maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell g open parentheses 3 x plus 1 close parentheses end cell row blank equals cell fraction numerator 2 open parentheses 3 x plus 1 close parentheses minus 1 over denominator open parentheses 3 x plus 1 close parentheses plus 3 end fraction end cell row blank equals cell fraction numerator 6 x plus 2 minus 1 over denominator 3 x plus 4 end fraction end cell row blank equals cell fraction numerator 6 x plus 1 over denominator 3 x plus 4 end fraction end cell end table end style 

Jadi begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 6 x plus 1 over denominator 3 x plus 4 end fraction end style, misalkan begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses equals y end style, maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell fraction numerator 6 x plus 1 over denominator 3 x plus 4 end fraction end cell row y equals cell fraction numerator 6 x plus 1 over denominator 3 x plus 4 end fraction end cell row cell y open parentheses 3 x plus 4 close parentheses end cell equals cell 6 x plus 1 end cell row cell 3 x y plus 4 y end cell equals cell 6 x plus 1 end cell row cell 3 x y minus 6 x end cell equals cell 1 minus 4 y end cell row cell x open parentheses 3 y minus 6 close parentheses end cell equals cell 1 minus 4 y end cell row x equals cell fraction numerator 1 minus 4 y over denominator 3 y minus 6 end fraction end cell row x equals cell fraction numerator negative 4 y plus 1 over denominator 3 y minus 6 end fraction end cell row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 4 x plus 1 over denominator 3 x minus 6 end fraction end cell end table end style  

Jadi, invers dari begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses end style adalah begin mathsize 14px style open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative 4 x plus 1 over denominator 3 x minus 6 end fraction end style, sehingga 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses negative 2 close parentheses end cell equals cell fraction numerator negative 4 open parentheses negative 2 close parentheses plus 1 over denominator 3 open parentheses negative 2 close parentheses minus 6 end fraction end cell row blank equals cell fraction numerator 8 plus 1 over denominator negative 6 minus 6 end fraction end cell row blank equals cell fraction numerator 9 over denominator negative 12 end fraction end cell row blank equals cell negative 3 over 4 end cell end table end style 

Jadi, jawaban yang tepat adalah C. 

0

Roboguru

Diketahui fungsi  dan , . Jika  dengan  adalah invers fungsi  dan  adalah invers fungsi , fungsi  ....

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals y row cell 2 x plus 4 end cell equals y row cell 2 x end cell equals cell y minus 4 end cell row x equals cell fraction numerator y minus 4 over denominator 2 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 x minus 4 over denominator 2 end fraction end cell end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis end cell equals cell g left parenthesis f to the power of negative 1 end exponent left parenthesis x right parenthesis right parenthesis end cell row blank equals cell g open parentheses fraction numerator x minus 4 over denominator 2 end fraction close parentheses end cell row blank equals cell fraction numerator 2 open parentheses begin display style fraction numerator x minus 4 over denominator 2 end fraction end style close parentheses plus 5 over denominator begin display style fraction numerator x minus 4 over denominator 2 end fraction minus 4 end style end fraction end cell row blank equals cell fraction numerator x minus 4 plus 5 over denominator begin display style fraction numerator x minus 4 minus 8 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator x plus 1 over denominator begin display style fraction numerator x minus 12 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator x plus 1 over denominator x minus 12 end fraction times 2 over 1 end cell row cell left parenthesis g ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x plus 2 over denominator x minus 12 end fraction end cell end table end style    

Karena begin mathsize 14px style h left parenthesis x right parenthesis equals left parenthesis g ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis end style maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x plus 2 over denominator x minus 12 end fraction end cell row cell h open parentheses x close parentheses end cell equals cell fraction numerator 2 x plus 2 over denominator x minus 12 end fraction end cell end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell h open parentheses x close parentheses end cell equals y row y equals cell fraction numerator 2 x plus 2 over denominator x minus 12 end fraction end cell row cell x y minus 12 y end cell equals cell 2 x plus 2 end cell row cell x y minus 2 x end cell equals cell 12 y plus 2 end cell row cell x open parentheses y minus 2 close parentheses end cell equals cell 12 y plus 2 end cell row x equals cell fraction numerator 12 y plus 2 over denominator y minus 2 end fraction end cell row cell h to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 12 x plus 2 over denominator x minus 2 end fraction end cell end table end style 

Jadi, jawaban yang tepat adalah C.

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Roboguru

Misalkan   dan  ,  tentukanlah :

Pembahasan Soal:

Diketahui:
begin mathsize 14px style straight f open parentheses straight x close parentheses equals fraction numerator 2 straight x plus 3 over denominator straight x minus 2 end fraction end style dan  begin mathsize 14px style straight g open parentheses straight x close parentheses equals square root of 5 straight x plus 3 end root end style,  

Ditanya : begin mathsize 14px style open parentheses straight f ring operator straight g close parentheses to the power of negative 1 end exponent open parentheses straight x close parentheses end style = ....

Penyelesaian :

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses straight f ring operator straight g close parentheses to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell open parentheses straight g to the power of negative 1 end exponent ring operator straight f to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses end cell end table end style 
 

kita cari invers dari undefined terlebih dahulu

maka untuk menentukan  begin mathsize 14px style straight f to the power of negative 1 end exponent open parentheses straight x close parentheses end style, kita misalkan undefined :
 

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

kita juga bisa gunakan cara cepat mencari invers fungsi

begin mathsize 14px style straight f open parentheses straight x close parentheses equals fraction numerator ax plus straight b over denominator cx plus straight d end fraction rightwards arrow straight f to the power of negative 1 end exponent open parentheses straight x close parentheses equals fraction numerator negative dx plus straight b over denominator cx minus straight a end fraction end style   
 

Jadi, dapat disimpulkan begin mathsize 14px style straight f open parentheses straight x close parentheses equals fraction numerator 2 straight x plus 3 over denominator straight x minus 2 end fraction rightwards arrow straight f to the power of negative 1 end exponent open parentheses straight x close parentheses equals fraction numerator 2 straight x plus 3 over denominator straight x minus 2 end fraction end style 

  

kemudian menentukan begin mathsize 14px style straight g to the power of negative 1 end exponent open parentheses straight x close parentheses end style, kita misalkan begin mathsize 14px style straight g to the power of negative 1 end exponent open parentheses straight x close parentheses equals straight y end style 
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row straight y equals cell square root of 5 straight x plus 3 end root end cell row cell straight y squared end cell equals cell 5 straight x plus 3 end cell row cell straight y squared minus 3 end cell equals cell 5 straight x end cell row cell fraction numerator straight y squared minus 3 over denominator 5 end fraction end cell equals straight x end table end style 
 

Jadi, dapat disimpulkan begin mathsize 14px style straight g to the power of negative 1 end exponent left parenthesis straight x right parenthesis equals fraction numerator straight x squared minus 3 over denominator 5 end fraction end style  

maka dengan menggunakan sifat invers komposisi fungsi

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses straight f ring operator straight g close parentheses to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell open parentheses straight g to the power of negative 1 end exponent ring operator straight f to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses end cell row blank equals cell fraction numerator open parentheses straight f open parentheses straight x close parentheses close parentheses squared minus 3 over denominator 5 end fraction end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator 2 straight x plus 3 over denominator straight x minus 2 end fraction end style close parentheses squared minus 3 over denominator 5 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 4 straight x squared plus 12 straight x plus 9 over denominator straight x squared minus 4 straight x plus 4 end fraction end style minus 3 over denominator 5 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 4 straight x squared plus 12 straight x plus 9 minus 3 open parentheses straight x squared minus 4 straight x plus 4 close parentheses over denominator straight x squared minus 4 straight x plus 4 end fraction end style over denominator 5 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 4 straight x squared plus 12 straight x plus 9 minus 3 straight x squared plus 12 straight x minus 12 over denominator straight x squared minus 4 straight x plus 4 end fraction end style over denominator 5 end fraction end cell row blank equals cell fraction numerator straight x squared plus 24 straight x minus 3 over denominator 5 open parentheses straight x squared minus 4 straight x plus 4 close parentheses end fraction end cell row blank equals cell fraction numerator straight x squared plus 24 straight x minus 3 over denominator 5 straight x squared minus 20 straight x plus 20 end fraction end cell end table end style 
 

Jadi, begin mathsize 14px style left parenthesis straight f ring operator straight g right parenthesis to the power of negative 1 end exponent left parenthesis straight x right parenthesis equals fraction numerator straight x squared plus 24 straight x minus 3 over denominator 5 straight x squared minus 20 straight x plus 20 end fraction end style 

0

Roboguru

Jika  dan , nilai dari

Pembahasan Soal:

Tentukan terlebih dahulu komposisi kedua fungsi tersebut, yaitu;

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell open parentheses f open parentheses g open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell f open parentheses 9 minus x close parentheses end cell row blank equals cell 2 open parentheses 9 minus x close parentheses plus 3 end cell row blank equals cell 18 minus 2 x plus 3 end cell row blank equals cell 21 minus 2 x end cell end table end style 

Kemudian tentukan invers dari fungsi komposisi yang telah diperoleh di atas, maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals y row cell 21 minus 2 x end cell equals y row cell negative 2 x end cell equals cell y minus 21 end cell row x equals cell fraction numerator y minus 21 over denominator negative 2 end fraction end cell row x equals cell fraction numerator negative y plus 21 over denominator 2 end fraction end cell end table end style 

Sehingga, invers fungsi komposisinya adalah begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative x plus 21 over denominator 2 end fraction end style .

 

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Roboguru

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