Roboguru

Diketahui , dan maka

Pertanyaan

Diketahui f left parenthesis x right parenthesis equals 2 x plus 5, dan g left parenthesis x right parenthesis equals fraction numerator negative 2 x over denominator x plus 4 end fraction maka open parentheses g 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 equals...

  1. fraction numerator 4 x minus 20 over denominator x minus 4 end fraction comma x not equal to 4

  2. fraction numerator 4 x minus 20 over denominator x plus 4 end fraction comma x not equal to negative 4

  3. fraction numerator 4 x plus 20 over denominator x minus 4 end fraction comma x not equal to 4

  4. fraction numerator 4 x plus 20 over denominator negative x plus 4 end fraction comma x not equal to 4

  5. fraction numerator negative 4 x minus 20 over denominator x minus 1 end fraction comma x not equal to 1

Pembahasan Soal:

Pertama kita tentukan masing masing invers dari setiap fungsi:

Ingat

f open parentheses x close parentheses equals a x plus b rightwards arrow f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus b over denominator a end fraction

f open parentheses x close parentheses equals fraction numerator a x plus b over denominator c x plus d end fraction rightwards arrow f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative d x plus b over denominator c x minus a end fraction

Sehingga

f left parenthesis x right parenthesis equals 2 x plus 5 rightwards arrow f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x minus 5 over denominator 2 end fraction  

g left parenthesis x right parenthesis equals fraction numerator negative 2 x over denominator x plus 4 end fraction rightwards arrow g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative 4 x over denominator x plus 2 end fraction

Maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g 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 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 blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator x minus 5 over denominator 2 end fraction close parentheses end cell row blank equals cell fraction numerator negative 4 open parentheses begin display style fraction numerator x minus 5 over denominator 2 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator x minus 5 over denominator 2 end fraction end style close parentheses plus 2 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative 4 x plus 20 over denominator 2 end fraction end style over denominator begin display style fraction numerator x minus 5 plus 4 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator negative 4 x plus 20 over denominator x minus 1 end fraction comma x not equal to 1 end cell end table 

Jadi, jawaban yang tepat adalah D

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Nasrullah

Mahasiswa/Alumni Universitas Negeri Makassar

Terakhir diupdate 06 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui . Jika , tentukanlah .

Pembahasan Soal:

Gunakan sifat invers fungsi komposisi dan rumus praktis invers fungsi berikut:

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 open parentheses g 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 f open parentheses x close parentheses equals a x plus b rightwards double arrow f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus b over denominator a end fraction end style  

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

begin mathsize 14px style g open parentheses x close parentheses equals 3 x plus 2 end style

Akan ditentukan 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.

Tentukan terlebih dahulu begin mathsize 14px style g to the power of negative 1 end exponent open parentheses x close parentheses end style dengan menggunakan rumus praktis, diperoleh:

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

Sehingga diperoleh 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 dengan perhitungan berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px 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 open parentheses g 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 row blank 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 blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator x plus 4 over denominator 2 x minus 3 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator x plus 4 over denominator 2 x minus 3 end fraction end style close parentheses minus 2 over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator x plus 4 over denominator 2 x minus 3 end fraction end style minus begin display style fraction numerator 2 open parentheses 2 x minus 3 close parentheses over denominator 2 x minus 3 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator x plus 4 minus 4 x plus 6 over denominator 2 x minus 3 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative 3 x plus 10 over denominator 2 x minus 3 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator negative 3 x plus 10 over denominator 2 x minus 3 end fraction cross times 1 third 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 negative 3 x plus 10 over denominator 6 x minus 9 end fraction end cell end table end style 

Jadi, dipeoleh 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 3 x plus 10 over denominator 6 x minus 9 end fraction end style.

Roboguru

dengan  dan  berturut-turut adalah invers dari fungsi  dan . Jika  maka

Pembahasan Soal:

Ingat sifat pada invers fungsi komposisi yaitu open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses left parenthesis x right parenthesis equals left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis, dari sifat tersebut maka akan didapatkan hasil yaitu sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses left parenthesis x right parenthesis end cell equals cell negative 2 x plus 4 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 negative 2 x plus 4 end cell row y equals cell negative 2 x plus 4 end cell row cell 2 x end cell equals cell 4 minus y end cell row x equals cell fraction numerator 4 minus y over denominator 2 end fraction end cell row cell open parentheses f ring operator g close parentheses left parenthesis y right parenthesis end cell equals cell fraction numerator 4 minus y over denominator 2 end fraction end cell row cell open parentheses f ring operator g close parentheses left parenthesis x right parenthesis end cell equals cell fraction numerator 4 minus x over denominator 2 end fraction end cell end table

Didapatkan hasil yaitu  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 4 minus x over denominator 2 end fraction end cell end table, kemudian menentukan fungsi g left parenthesis x right parenthesis dengan memanfaatkan definisi dari komposisi fungsi.

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 4 minus x over denominator 2 end fraction end cell row cell f left square bracket g left parenthesis x right parenthesis right square bracket end cell equals cell fraction numerator 4 minus x over denominator 2 end fraction end cell row cell fraction numerator negative 2 open parentheses g open parentheses x close parentheses close parentheses plus 4 over denominator 2 open parentheses g left parenthesis x close parentheses right parenthesis minus 10 end fraction end cell equals cell fraction numerator 4 minus x over denominator 2 end fraction end cell row cell 2 open square brackets negative 2 open parentheses g open parentheses x close parentheses close parentheses plus 4 close square brackets end cell equals cell open square brackets 2 open parentheses g left parenthesis x close parentheses right parenthesis minus 10 close square brackets open square brackets 4 minus x close square brackets end cell row cell negative 4 g left parenthesis x right parenthesis plus 8 end cell equals cell 8 g left parenthesis x right parenthesis minus 40 minus 2 x g left parenthesis x right parenthesis plus 10 x end cell row cell negative 4 g left parenthesis x right parenthesis plus 8 end cell equals cell 8 g left parenthesis x right parenthesis minus 40 minus 2 x g left parenthesis x right parenthesis plus 10 x end cell row cell negative 4 g left parenthesis x right parenthesis minus 8 g left parenthesis x right parenthesis plus 2 x g left parenthesis x right parenthesis end cell equals cell negative 40 plus 10 x minus 8 end cell row cell g left parenthesis x right parenthesis left parenthesis negative 4 minus 8 plus 2 x right parenthesis end cell equals cell 10 x minus 48 end cell row cell g left parenthesis x right parenthesis left parenthesis negative 12 plus 2 x right parenthesis end cell equals cell 10 x minus 48 end cell row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 10 x minus 48 over denominator 12 plus 2 x end fraction end cell row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x minus 24 over denominator 6 plus x end fraction end cell end table

Didapatkan  table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x minus 24 over denominator 6 plus x end fraction end cell end table, maka nilai dari g left parenthesis 6 right parenthesis adalah 

table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x minus 24 over denominator 6 plus x end fraction end cell row cell g left parenthesis 6 right parenthesis end cell equals cell fraction numerator 5 left parenthesis 6 right parenthesis minus 24 over denominator 6 plus 6 end fraction end cell row blank equals cell fraction numerator 30 minus 24 over denominator 12 end fraction end cell row blank equals cell 6 over 12 end cell row blank equals cell 1 half end cell end table

Jadi, dapat disimpulkan bahwa nilai dari dari g left parenthesis 6 right parenthesis adalah 1 half.

Oleh karena itu, tidak ada jawaban yang tepat.

Roboguru

2. Fungsi  didefinisikan oleh  dan . Tentukan: a.

Pembahasan Soal:

Perhatikan penghitungan berikut!

 f open parentheses x close parentheses equals 2 x plus 11 misal colon space y equals f open parentheses x close parentheses space maka space x equals f to the power of negative 1 end exponent open parentheses x close parentheses f open parentheses x close parentheses equals 2 x plus 11 y equals 2 x plus 11 y minus 11 equals 2 x fraction numerator y minus 11 over denominator 2 end fraction equals x f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus 11 over denominator 2 end fraction  g open parentheses x close parentheses equals 1 minus x misal colon space z equals g open parentheses x close parentheses space maka space x equals g to the power of negative 1 end exponent open parentheses x close parentheses g open parentheses x close parentheses equals 1 minus x z equals 1 minus x x equals 1 minus z g to the power of negative 1 end exponent open parentheses x close parentheses equals 1 minus x  

sehingga, open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses yaitu:

 open parentheses f 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 f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses open parentheses f 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 open parentheses f to the power of negative 1 end exponent open parentheses x close parentheses close parentheses close parentheses open parentheses f 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 open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses close parentheses open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals 1 minus open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals 2 over 2 minus open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 2 minus x plus 11 over denominator 2 end fraction open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 13 minus x over denominator 2 end fraction           

Jadi, open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 13 minus x over denominator 2 end fraction .space 

Roboguru

Diketahui fungsi  dan fungsi inversnya , maka nilai untuk  adalah ....

Pembahasan Soal:

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

Diketahui bahwa begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses equals 3 x end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell 3 x end cell row cell fraction numerator 2 x plus 3 over denominator 4 minus x end fraction end cell equals cell 3 x end cell row cell 2 x plus 3 end cell equals cell 3 x open parentheses 4 minus x close parentheses end cell row cell 2 x plus 3 end cell equals cell 12 x minus 3 x squared end cell row cell 3 x squared minus 10 x plus 3 end cell equals 0 row cell open parentheses 3 x minus 1 close parentheses open parentheses x minus 3 close parentheses end cell equals 0 end table end style 

begin mathsize 14px style 3 x subscript 1 minus 1 equals 0 3 x subscript 1 equals 1 x subscript 1 equals 1 third end style   atau   begin mathsize 14px style x subscript 2 minus 3 equals 0 x subscript 2 equals 3 end style 

Maka, pilihan jawaban yang benar adalah B.

Roboguru

Diketahui  dan . b. Apakah. c. Sifat apakah yang dapat kalian temukan dari fungsi invers pada fungsi komposisi !

Pembahasan Soal:

Diketahui

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell x plus 2 end cell row cell g open parentheses x close parentheses end cell equals cell 3 x minus 10 end cell end table

Dengan menggunakan konsep invers dan komposisi fungsi diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals y row cell x plus 2 end cell equals y row x equals cell y minus 2 end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell x minus 2 end cell end table  dan   table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses x close parentheses end cell equals y row cell 3 x minus 10 end cell equals y row cell 3 x end cell equals cell y plus 10 end cell row x equals cell fraction numerator y plus 10 over denominator 3 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 plus 10 over denominator 3 end fraction end cell end table

b. Akan ditunjukkan open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses

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 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 g 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 fraction numerator x plus 10 over denominator 3 end fraction close parentheses end cell row blank equals cell fraction numerator x plus 10 over denominator 3 end fraction minus 2 end cell row blank equals cell fraction numerator x plus 10 minus 6 over denominator 3 end fraction end cell row blank equals cell fraction numerator x plus 4 over denominator 3 end fraction end cell end table

Disisi lain

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 x plus 2 close parentheses end cell row blank equals cell 3 open parentheses x plus 2 close parentheses minus 10 end cell row blank equals cell 3 x plus 6 minus 10 end cell row blank equals cell 3 x minus 4 end cell row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals y row cell 3 x minus 4 end cell equals y row cell 3 x end cell equals cell y plus 4 end cell row x equals cell fraction numerator y plus 4 over denominator 3 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 x plus 4 over denominator 3 end fraction end cell end table

Dengan demikian terbukti bahwa open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses.

c. Sehingga diperoleh sifat invers komposisi fungsi ssebagai berikut

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 to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell equals cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell row cell open parentheses g 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 open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell end table.

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