Roboguru

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

Pertanyaan

Diketahui f left parenthesis x right parenthesis equals fraction numerator x minus 6 over denominator 2 x plus 1 end fractionx not equal to negative 1 half, dan g left parenthesis x right parenthesis equals x minus 3. Jika f to the power of negative 1 end exponent adalah invers dari f, maka open parentheses f to the power of negative 1 end exponent ring operator g close parentheses open parentheses x close parentheses adalah...space

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

  2. fraction numerator x plus 3 over denominator 6 minus 2 x end fraction comma space x not equal to 3

  3. fraction numerator x plus 3 over denominator 7 minus 2 x end fraction comma space x not equal to 7 over 2

  4. fraction numerator x plus 6 over denominator 1 minus 2 x end fraction comma space x not equal to 1 half

  5. fraction numerator x plus 6 over denominator 7 minus 2 x end fraction comma space x not equal to 7 over 2

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell fraction numerator x minus 6 over denominator 2 x plus 1 end fraction end cell row y equals cell fraction numerator x minus 6 over denominator 2 x plus 1 end fraction end cell row cell y open parentheses 2 x plus 1 close parentheses end cell equals cell x minus 6 end cell row cell 2 x y plus y end cell equals cell x minus 6 end cell row cell y plus 6 end cell equals cell x minus 2 x y end cell row cell y plus 6 end cell equals cell x left parenthesis 1 minus 2 y right parenthesis end cell row x equals cell fraction numerator y plus 6 over denominator 1 minus 2 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 x plus 6 over denominator 1 minus 2 x end fraction end cell end table

 

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f to the power of negative 1 end exponent ring operator g close parentheses open parentheses x close parentheses end cell equals cell f to the power of negative 1 end exponent open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell fraction numerator open parentheses open parentheses x minus 3 close parentheses plus 6 close parentheses over denominator open parentheses 1 minus 2 open parentheses x minus 3 close parentheses close parentheses end fraction end cell row blank equals cell fraction numerator x plus 3 over denominator 1 minus 2 x plus 6 end fraction end cell row blank equals cell fraction numerator x plus 3 over denominator 7 minus 2 x end fraction end cell end table
 

Oleh karena itu, jawaban yang benar adalah C.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

G. Albiah

Mahasiswa/Alumni Universitas Galuh Ciamis

Terakhir diupdate 01 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Fungsi  dan  berturut-turut didefinisikan  dan , rumus fungsi invers  adalah

Pembahasan Soal:

Diketahui f open parentheses x close parentheses equals fraction numerator x plus 4 over denominator x minus 6 end fraction comma space x not equal to 6 dan g open parentheses x close parentheses equals 2 x minus 1, maka

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 f open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell f open parentheses 2 x minus 1 close parentheses end cell row blank equals cell fraction numerator 2 x minus 1 plus 4 over denominator 2 x minus 1 minus 6 end fraction end cell row blank equals cell fraction numerator 2 x plus 3 over denominator 2 x minus 7 end fraction end cell end table 

Inversnya adalah

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator 2 x plus 3 over denominator 2 x minus 7 end fraction end cell row cell y open parentheses 2 x minus 7 close parentheses end cell equals cell 2 x plus 3 end cell row cell 2 x y minus 7 y end cell equals cell 2 x plus 3 end cell row cell 2 x y minus 2 x end cell equals cell 7 y plus 3 end cell row cell x left parenthesis 2 y minus 2 right parenthesis end cell equals cell 7 y plus 3 end cell row x equals cell fraction numerator 7 y plus 3 over denominator 2 y minus 2 end fraction end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 7 x plus 3 over denominator 2 x minus 2 end fraction comma space x not equal to 1 end cell end table 

Jadi, jawaban yang tepat adalah E.

Roboguru

Ditentukan oleh  dan  maka  adalah ....

Pembahasan Soal:

Fungsi komposisi undefined didapatkan:

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 f open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell f open parentheses x plus 2 close parentheses end cell row blank equals cell 2 open parentheses x plus 2 close parentheses plus 5 end cell row blank equals cell 2 x plus 4 plus 5 end cell row blank equals cell 2 x plus 9 end cell end table end style 

Maka 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 end style didapatkan:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 2 x plus 9 end cell row cell y minus 9 end cell equals cell 2 x end cell row cell fraction numerator y minus 9 over denominator 2 end fraction end cell equals x row x equals cell fraction numerator y minus 9 over denominator 2 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 x minus 9 over denominator 2 end fraction end cell end table end style   

Jadi, 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 end style adalah begin mathsize 14px style fraction numerator x minus 9 over denominator 2 end fraction end style.

Roboguru

Diketahui fungsi sebagai berikut:   Tentukan: b.  dan  h.  i.   j. Apakah . Apakah .

Pembahasan Soal:

Gunakan konsep komposisi fungsi dan invers fungsi.

Diketahui:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell f open parentheses x close parentheses end cell equals cell 2 x plus 1 end cell row cell h open parentheses x close parentheses end cell equals cell x minus 4 end cell row cell g open parentheses x close parentheses end cell equals cell x squared minus 3 x plus 2 end cell row cell m open parentheses x close parentheses end cell equals cell fraction numerator x plus 4 over denominator 2 x minus 5 end fraction end cell end table end style

b. Menentukan begin mathsize 14px style open parentheses h ring operator f close parentheses open parentheses x close parentheses end style dan begin mathsize 14px style open parentheses f ring operator h close parentheses open parentheses x close parentheses end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses h ring operator f close parentheses open parentheses x close parentheses end cell equals cell h open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell h open parentheses 2 x plus 1 close parentheses end cell row blank equals cell open parentheses 2 x plus 1 close parentheses minus 4 end cell row cell open parentheses h ring operator f close parentheses open parentheses x close parentheses end cell equals cell 2 x minus 3 end cell end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses f ring operator h close parentheses open parentheses x close parentheses end cell equals cell f open parentheses h open parentheses x close parentheses close parentheses end cell row blank equals cell f open parentheses x minus 4 close parentheses end cell row blank equals cell 2 open parentheses x minus 4 close parentheses plus 1 end cell row blank equals cell 2 x minus 8 plus 1 end cell row cell open parentheses f ring operator h close parentheses open parentheses x close parentheses end cell equals cell 2 x minus 7 end cell end table end style 

Diperoleh begin mathsize 14px style open parentheses h ring operator f close parentheses open parentheses x close parentheses equals 2 x minus 3 end style dan begin mathsize 14px style open parentheses f ring operator h close parentheses open parentheses x close parentheses equals 2 x minus 7 end style.

h. menentukan begin mathsize 14px style open parentheses h ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style

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

i. Menentukan begin mathsize 14px style open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end style

Terlebih dahulu tentukan begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses end style dan begin mathsize 14px style h to the power of negative 1 end exponent open parentheses x close parentheses end style.

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

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

Sehingga diperoleh begin mathsize 14px style open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end style sebagai berikut:

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

j. Apakah begin mathsize 14px style open parentheses h ring operator f close parentheses open parentheses x close parentheses equals open parentheses f ring operator h close parentheses open parentheses x close parentheses end style. Apakah begin mathsize 14px style open parentheses h ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end style.

Jika diperhatikan pada perhitungan di atas diperoleh:

begin mathsize 14px style open parentheses h ring operator f close parentheses open parentheses x close parentheses not equal to open parentheses f ring operator h close parentheses open parentheses x close parentheses end style 

dan begin mathsize 14px style open parentheses h ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses not equal to open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end style 

Roboguru

Diketahui  . Jika   , maka  ....

Pembahasan Soal:

Kita akan lebih mudah mengerjakannya dengan sifat berikut begin mathsize 14px style open parentheses f ring operator h ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses g to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end style 

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

Jadi, jawabannya B.

Roboguru

Diketahui fungsi  dan , Fungsi  adalah ...

Pembahasan Soal:

Pertama akan dicari invers dari fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 2 x over denominator x plus 3 end fraction space comma space x not equal to negative 3 end style

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

selanjutnya akan dicari invers dari fungsi begin mathsize 14px style g left parenthesis x right parenthesis equals fraction numerator x plus 1 over denominator x end fraction comma space x not equal to 0 end style

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

Dengan menggunakan sifat invers akan ditentukan  begin mathsize 14px style left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent open parentheses x close parentheses end style  

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

Nilai fugsi begin mathsize 14px style left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent open parentheses 1 third close parentheses end style adalah

begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 1 third close parentheses equals fraction numerator 2 minus begin display style 1 third end style over denominator 4 open parentheses begin display style 1 third end style close parentheses minus 2 end fraction space space space space space space space space space space space space space space space space space space equals fraction numerator begin display style 5 over 3 end style over denominator begin display style fraction numerator 4 minus 6 over denominator 3 end fraction end style end fraction space space space space space space space space space space space space space space space space space space equals 5 over 3 times 3 over 2 space space space space space space space space space space space space space space space space space space equals 5 over 2 end style 

Jadi,jawaban yang tepat adalah D

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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