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Jika fungsi f dan g memiliki invers dan memenuhi f(2x)=g(x−5), rumus invers fungsi  adalah f−1(x)=....

Pertanyaan

 Jika fungsi undefined dan undefined memiliki invers dan memenuhi begin mathsize 14px style f left parenthesis 2 x right parenthesis equals g left parenthesis x minus 5 right parenthesis end style, rumus invers fungsi undefined adalah begin mathsize 14px style f to the power of negative 1 end exponent left parenthesis x right parenthesis equals end style....

  1. begin mathsize 14px style g to the power of negative 1 end exponent left parenthesis x right parenthesis plus 5 end style 

  2. begin mathsize 14px style g to the power of negative 1 end exponent left parenthesis x right parenthesis minus 10 end style 

  3. begin mathsize 14px style 2 g to the power of negative 1 end exponent left parenthesis x right parenthesis plus 5 end style 

  4. begin mathsize 14px style 2 g to the power of negative 1 end exponent left parenthesis x right parenthesis minus 10 end style 

  5. begin mathsize 14px style 2 g to the power of negative 1 end exponent left parenthesis x right parenthesis plus 10 end style 

Pembahasan Soal:

Diketahui: begin mathsize 14px style f left parenthesis 2 x right parenthesis equals g left parenthesis x minus 5 right parenthesis end style maka 

begin mathsize 14px style 2 x equals f to the power of negative 1 end exponent times g open parentheses x minus 5 close parentheses end style 

Misalkan begin mathsize 14px style g open parentheses x minus 5 close parentheses equals a end style maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses x minus 5 close parentheses end cell equals a row cell x minus 5 end cell equals cell g to the power of negative 1 end exponent open parentheses a close parentheses end cell row x equals cell g to the power of negative 1 end exponent open parentheses a close parentheses plus 5 end cell end table end style 

Sehingga diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 open parentheses g to the power of negative 1 end exponent open parentheses a close parentheses plus 5 close parentheses end cell equals cell f to the power of negative 1 end exponent open parentheses a close parentheses end cell row cell 2 times g to the power of negative 1 end exponent open parentheses a close parentheses plus 10 end cell equals cell f to the power of negative 1 end exponent open parentheses a close parentheses end cell row cell 2 times g to the power of negative 1 end exponent open parentheses x close parentheses plus 10 end cell equals cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell end table end style 

Jadi, jawaban yang tepat adalah E.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Amamah

Mahasiswa/Alumni Universitas Negeri Malang

Terakhir diupdate 05 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Invers dari fungsi f(x)=2x−54x+3​ adalah...

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 4 x plus 3 over denominator 2 x minus 5 end fraction end cell row y equals cell fraction numerator 4 x plus 3 over denominator 2 x minus 5 end fraction end cell row cell y apostrophe left parenthesis 2 x minus 5 right parenthesis end cell equals cell 4 x plus 3 end cell row cell 2 x y apostrophe minus 5 y apostrophe end cell equals cell 4 x plus 3 end cell row cell 2 x y apostrophe minus 5 y apostrophe minus 4 x minus 3 end cell equals 0 row cell 2 x y apostrophe minus 4 x minus 5 y apostrophe minus 3 end cell equals 0 row cell x left parenthesis 2 y apostrophe minus 4 right parenthesis minus 5 y apostrophe minus 3 end cell equals 0 row cell x left parenthesis 2 y apostrophe minus 4 right parenthesis end cell equals cell 5 y apostrophe plus 3 end cell row x equals cell fraction numerator 5 y apostrophe plus 3 over denominator 2 y apostrophe 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 5 x plus 3 over denominator 2 x minus 4 end fraction end cell end table

Jadi, Invers dari fungsi f left parenthesis x right parenthesis equals fraction numerator 4 x plus 3 over denominator 2 x minus 5 end fraction 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 5 x plus 3 over denominator 2 x minus 4 end fraction end cell end table.

0

Roboguru

Diketahui f(x)=2x–1dang(x)=3x+5. Tentukan (f∘g)−1(x)!

Pembahasan Soal:

Dari soal tersebut, pertama-tama tentukan nilai dari begin mathsize 14px style left parenthesis f ring operator g right parenthesis space left parenthesis x right parenthesis end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell 2 left parenthesis 3 x plus 5 right parenthesis minus 1 end cell row blank equals cell 6 x plus 9 end cell end table end style

Setelah itu, tentukan fungsi inversnya.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell 6 x plus 9 end cell row y equals cell 6 x plus 9 end cell row cell 6 x end cell equals cell y minus 9 end cell row x equals cell fraction numerator y minus 9 over denominator 6 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 6 end fraction end cell end table end style

Jadi, nilai table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis f ring operator g right parenthesis 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 x minus 9 over denominator 6 end fraction end cell end table.

0

Roboguru

f(x)=x−4,g(x)=x−511x+4​,x=5  Tentukan (f∘g)−1(x).

Pembahasan Soal:

open parentheses f ring operator g close parentheses open parentheses x close parentheses equals f open parentheses g open parentheses x close parentheses close parentheses space space space space space space space space space space space space space equals f open parentheses fraction numerator 11 x plus 4 over denominator x minus 5 end fraction close parentheses space space space space space space space space space space space space space equals fraction numerator 11 x plus 4 over denominator x minus 5 end fraction minus 4 space space space space space space space space space space space space space equals fraction numerator open parentheses 11 x plus 4 close parentheses minus 4 open parentheses x minus 5 close parentheses over denominator x minus 5 end fraction space space space space space space space space space space space space space equals fraction numerator 7 x plus 24 over denominator x minus 5 end fraction 

sekarang kita akan mencari invers fungsinya,

misal space fraction numerator 7 x plus 24 over denominator x minus 5 end fraction equals a space space space space space space space space space space 7 x plus 24 equals a open parentheses x minus 5 close parentheses space space space space space space space space space space 7 x plus 24 equals a x minus 5 a space space space space space space space space space space 7 x minus a x equals negative 5 a minus 24 space space space space space space space space space space open parentheses 7 minus a close parentheses x equals negative 5 a minus 24 space space space space space space space space space space space space space space space space space space space space x equals fraction numerator negative 5 a minus 24 over denominator 7 minus a end fraction space space space space space space space space space space space space space space space space space space space space x equals fraction numerator 5 a plus 24 over denominator a minus 7 end fraction space space space space space space space space space space space space rightwards arrow space space space space space open parentheses f ring operator g close parentheses to the power of negative 1 end exponent equals fraction numerator 5 x plus 24 over denominator x minus 7 end fraction 

Jadi, open parentheses f ring operator g close parentheses to the power of negative 1 end exponent equals fraction numerator 5 x plus 24 over denominator x minus 7 end fraction.

0

Roboguru

Diketahui fungsi f:R→R dan g:R→R dengan f(x)=x3x+5​ dan g(x)=2x−1, maka (f∘g)−1(x)=…

Pembahasan Soal:

Diketahui fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 3 x plus 5 over denominator x end fraction end style dan begin mathsize 14px style g left parenthesis x right parenthesis equals 2 x minus 1 end style. Komposisi fungsi undefined dapat dihitung seperti berikut,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell row blank equals cell f left parenthesis 2 x minus 1 right parenthesis end cell row blank equals cell fraction numerator 3 left parenthesis 2 x minus 1 right parenthesis plus 5 over denominator 2 x minus 1 end fraction end cell row blank equals cell fraction numerator 6 x minus 3 plus 5 over denominator 2 x minus 1 end fraction end cell row blank equals cell fraction numerator 6 x plus 2 over denominator 2 x minus 1 end fraction comma space x not equal to 1 half end cell end table end style 

Bentuk umum invers fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction end style adalah

begin mathsize 14px style f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative d x plus b over denominator c x minus a end fraction comma space x not equal to a over c end style.

Berdasarkan rumus invers di atas, bentuk invers untuk fungsi begin mathsize 14px style open parentheses f ring operator g close parentheses left parenthesis x right parenthesis equals fraction numerator 6 x plus 2 over denominator 2 x minus 1 end fraction end style adalah

begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative left parenthesis negative 1 right parenthesis x plus 2 over denominator 2 x minus 6 end fraction equals fraction numerator x plus 2 over denominator 2 x minus 6 end fraction comma space x not equal to 3 end style 

Dengan demikian, fungsi invers begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis end style adalah fraction numerator size 14px x size 14px plus size 14px 2 over denominator size 14px 2 size 14px x size 14px minus size 14px 6 end fraction.

Jadi, jawaban yang tepat adalah E. 

0

Roboguru

Diketahui f(x)=2x+5 dan g(x)=x−4. Invers dari f(x) adalah f−1(x) dan invers g(x) adalah g−1(x), maka (g−1∘f−1)(x) adalah …

Pembahasan Soal:

Ingatlah sifat invers fungsi komposisi:

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

Oleh karena itu, kita akan menentukan fungsi komposisi open parentheses f ring operator g close parentheses open parentheses x close parentheses terlebih dahulu.

Diketahui f open parentheses x close parentheses equals 2 x plus 5 dan g open parentheses x close parentheses equals x minus 4. 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 x minus 4 close parentheses end cell row blank equals cell 2 open parentheses x minus 4 close parentheses plus 5 end cell row blank equals cell 2 x minus 8 plus 5 end cell row blank equals cell 2 x minus 3 end cell end table

Selanjutnya, misalkan y equals open parentheses f ring operator g close parentheses open parentheses x close parentheses. Maka

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

Dengan demikian, open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis y right parenthesis equals 1 half open parentheses y plus 3 close parentheses sehingga open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals 1 half open parentheses x plus 3 close parentheses dan mengakibatkan

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 half open parentheses x plus 3 close parentheses

Jadi, jawaban yang tepat adalah E.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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