Roboguru

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

Pertanyaan

f open parentheses x close parentheses equals x minus 4 comma space g open parentheses x close parentheses equals fraction numerator 11 x plus 4 over denominator x minus 5 end fraction comma space x not equal to 5 

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

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.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

L. Marlina

Mahasiswa/Alumni Institut Teknologi Bandung

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui: f(x)=5x+7g(x)=−2x+1h(x)=7x+1  Tentukan ((f∘g)∘h)−1(x)=…

Pembahasan Soal:

Diketahui:

begin mathsize 14px style f open parentheses x close parentheses equals 5 x plus 7 g open parentheses x close parentheses equals negative 2 x plus 1 h open parentheses x close parentheses equals 7 x plus 1 end style

Langkah-langkah mencari begin mathsize 14px style left parenthesis open parentheses f ring operator g close parentheses ring operator h right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end style sebagai berikut:

1. mencari begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style dengan memasukkan begin mathsize 14px style g open parentheses x close parentheses end style ke dalam fungsi begin mathsize 14px style f open parentheses x close parentheses end style, 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 f open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell f open parentheses negative 2 x plus 1 close parentheses end cell row blank equals cell 5 open parentheses negative 2 x plus 1 close parentheses plus 7 end cell row blank equals cell negative 10 x plus 5 plus 7 end cell row blank equals cell negative 10 x plus 12 end cell end table end style 

2. mencari begin mathsize 14px style open parentheses open parentheses f ring operator g close parentheses ring operator h close parentheses open parentheses x close parentheses end style dengan memasukkan begin mathsize 14px style h open parentheses x close parentheses end style ke dalam fungsi begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style, yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses open parentheses f ring operator g close parentheses ring operator h close parentheses open parentheses x close parentheses end cell equals cell open parentheses f ring operator g close parentheses open parentheses h open parentheses x close parentheses close parentheses end cell row blank equals cell open parentheses f ring operator g close parentheses open parentheses 7 x plus 1 close parentheses end cell row blank equals cell negative 10 open parentheses 7 x plus 1 close parentheses plus 12 end cell row blank equals cell negative 70 x minus 10 plus 12 end cell row blank equals cell negative 70 x plus 2 end cell row blank blank blank end table end style 

3. mencari begin mathsize 14px style left parenthesis open parentheses f ring operator g close parentheses ring operator h right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end style, yaitu

misalkan begin mathsize 14px style y equals left parenthesis left parenthesis f ring operator g right parenthesis ring operator h 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 open parentheses open parentheses f ring operator g close parentheses ring operator h close parentheses open parentheses x close parentheses end cell equals cell negative 70 x plus 2 end cell row y equals cell negative 70 x plus 2 end cell row cell negative 70 x end cell equals cell y minus 2 end cell row x equals cell fraction numerator y minus 2 over denominator negative 70 end fraction end cell row cell open parentheses open parentheses f ring operator g close parentheses ring operator h close parentheses to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator y minus 2 over denominator negative 70 end fraction end cell row cell open parentheses open parentheses f ring operator g close parentheses ring operator h 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 negative 70 end fraction end cell end table end style 

Oleh karena itu, jawaban yang tepat adalah D.

0

Roboguru

Diketahui fungsi f(x)=2x+1 dan g(x)=x+3. Tentukan : d. (g−1∘f)(1)

Pembahasan Soal:

begin mathsize 14px style g left parenthesis x right parenthesis equals x plus 3 y equals x plus 3 x equals y minus 3  g to the power of negative 1 end exponent left parenthesis x right parenthesis equals x minus 3  open parentheses g to the power of negative 1 end exponent ring operator f close parentheses open parentheses 1 close parentheses equals g to the power of negative 1 end exponent open parentheses f open parentheses 1 close parentheses close parentheses equals g to the power of negative 1 end exponent left parenthesis 2 left parenthesis 1 right parenthesis plus 1 right parenthesis equals g to the power of negative 1 end exponent left parenthesis 3 right parenthesis equals x minus 3 equals 3 minus 3 equals 0 end style

0

Roboguru

Jika f dan g adalah fungsi dalam R dengan f(x)=x+5 dan g(x)=3x+4, tentukan: b. (g∘f)−1(x)

Pembahasan Soal:

Diketahui begin mathsize 14px style f open parentheses x close parentheses equals x plus 5 end style dan begin mathsize 14px style g open parentheses x close parentheses equals 3 x plus 4 end style

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 3 open parentheses x plus 5 close parentheses plus 4 end cell row blank equals cell 3 x plus 19 end cell end table end style

Selanjutnya, akan ditentukan 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 end style sebagai berikut.

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Dengan demikian, Error converting from MathML to accessible text. 

0

Roboguru

Jika f(x)=5x−2 dan g(x)=4x−3, maka (f−1∘g−1)(x)=....

Pembahasan Soal:

Jika dan ,

menggunakan konsep invers fungsi komposisi :

size 14px left parenthesis size 14px f to the power of size 14px minus size 14px 1 end exponent size 14px ring operator size 14px g to the power of size 14px minus size 14px 1 end exponent size 14px right parenthesis begin mathsize 14px style left parenthesis x right parenthesis end style size 14px equals begin mathsize 14px style left parenthesis g ring operator f right parenthesis end style to the power of size 14px minus size 14px 1 end exponent size 14px left parenthesis size 14px x size 14px right parenthesis

maka kita cari dulu fungsi komposisinya:

begin mathsize 14px style open parentheses g ring operator f close parentheses left parenthesis x right parenthesis equals g left parenthesis f left parenthesis x right parenthesis right parenthesis end style

                begin mathsize 14px style equals g left parenthesis 5 x minus 2 right parenthesis equals 4 left parenthesis 5 x minus 2 right parenthesis minus 3 equals 20 x minus 8 minus 3 equals 20 x minus 11 end style

sedemikian sehingga inversnya adalah :

size 14px y size 14px equals size 14px 20 size 14px x size 14px minus size 14px 11 size 14px y size 14px plus size 14px 11 size 14px equals size 14px 20 size 14px x fraction numerator size 14px y size 14px plus size 14px 11 over denominator size 14px 20 end fraction size 14px equals size 14px x left parenthesis size 14px g size 14px ring operator size 14px f size 14px right parenthesis to the power of size 14px minus size 14px 1 end exponent size 14px left parenthesis size 14px x size 14px right parenthesis size 14px equals fraction numerator size 14px x size 14px plus size 14px 11 over denominator size 14px 20 end fraction 

jadi, jawabannya E

0

Roboguru

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

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.

1

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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