Roboguru

Invers fungsi dari f(x)=32​x−65​ adalah f−1(x)= ...

Pertanyaan

Invers fungsi dari straight f left parenthesis x right parenthesis equals 2 over 3 x minus 5 over 6 adalah straight f to the power of negative 1 end exponent left parenthesis x right parenthesis equals ...

Pembahasan Soal:

Untuk menjawab soal, digunakan konsep mencari fungsi invers dari fungsi linear.

Karena begin mathsize 14px style y equals text f end text left parenthesis x right parenthesis end style, maka y equals 2 over 3 x minus 5 over 6.

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 2 over 3 x minus 5 over 6 end cell row cell 2 over 3 x end cell equals cell y plus 5 over 6 end cell row x equals cell fraction numerator y plus begin display style 5 over 6 end style over denominator begin display style 2 over 3 end style end fraction end cell row x equals cell fraction numerator y over denominator begin display style 2 over 3 end style end fraction plus fraction numerator begin display style 5 over 6 end style over denominator begin display style 2 over 3 end style end fraction end cell row x equals cell 3 over 2 y plus 5 over 4 end cell end table
 

karena begin mathsize 14px style x equals f to the power of negative 1 end exponent open parentheses y close parentheses end style, maka f to the power of negative 1 end exponent left parenthesis y right parenthesis equals 3 over 2 y plus 5 over 4, Sehingga f to the power of negative 1 end exponent left parenthesis x right parenthesis equals 3 over 2 x plus 5 over 4.
 

Jadi, fungsi invers dari f left parenthesis x right parenthesis equals 2 over 3 x minus 5 over 6 adalah f to the power of negative 1 end exponent left parenthesis x right parenthesis equals 3 over 2 x plus 5 over 4.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

D. Nuryani

Mahasiswa/Alumni Universitas Padjadjaran

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Diketahui fungsi f(x)=2−x+5​, Inversnya adalah f−1(x)=...

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell 2 minus square root of x plus 5 end root end cell row y equals cell 2 minus square root of x plus 5 end root end cell row cell y minus 2 end cell equals cell square root of x plus 5 end root end cell row cell open parentheses y minus 2 close parentheses squared end cell equals cell x plus 5 end cell row x equals cell open parentheses y minus 2 close parentheses squared minus 5 end cell row x equals cell y squared minus 4 y plus 4 minus 5 end cell row x equals cell y squared minus 4 y minus 1 end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell x squared minus 4 x minus 1 end cell end table

Oleh karena itu, jawaban yang benar adalah A.

0

Roboguru

Untuk no. 1-5 tentukan invers fungsi di bawah ini 3. f(x)=−21​x+4

Pembahasan Soal:

Perhatikan penghitungan berikut!

begin mathsize 14px style misalkan space y equals f left parenthesis x right parenthesis comma space maka space x equals f to the power of negative 1 end exponent left parenthesis x right parenthesis f left parenthesis x right parenthesis space space space space space space space space space equals negative 1 half x plus 4 y space space space space space space space space space space space space space equals negative 1 half x plus 4 y minus 4 space space space space space space space space equals negative 1 half x minus 2 left parenthesis y minus 4 right parenthesis equals x minus 2 y plus 8 space space space equals x sehingga comma f to the power of negative 1 end exponent left parenthesis x right parenthesis equals negative 2 x plus 8 end style 

Jadi, invers fungsi tersebut adalah begin mathsize 14px style negative 2 x plus 8. end style 

0

Roboguru

Diketahui fungsi f dan g adalah fungsi bijektif yang ditentukan dengan f(x)=2x+5 dan g(x)=x−2. Tentukan : b. f−1 dan g−1  c. (g−1∘f−1) dan (f−1∘g−1)

Pembahasan Soal:

Bentuk umum fungsi invers dari fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals a x plus b end style adalah begin mathsize 14px style f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x minus b over denominator a end fraction end style.

Diberikan fungsi-fungsi bijektif  undefined dengan begin mathsize 14px style a equals 2 end style dan begin mathsize 14px style b equals 5 end style, serta fungsi begin mathsize 14px style g left parenthesis x right parenthesis equals x minus 2 end style dengan begin mathsize 14px style a equals 1 end style dan begin mathsize 14px style b equals negative 2 end style.

Dengan demikian, invers dari kedua fungsi linear tersebut dapat dituliskan seperti berikut,

begin mathsize 14px style f to the power of negative 1 end exponent equals fraction numerator x minus 5 over denominator 2 end fraction end style dan begin mathsize 14px style g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x minus 1 over denominator negative 2 end fraction equals fraction numerator 1 minus x over denominator 2 end fraction end style.

Komposisi dari kedua fungsi invers di atas adalah

begin mathsize 14px style 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 g to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent left parenthesis x right parenthesis 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 1 minus open parentheses begin display style fraction numerator x minus 5 over denominator 2 end fraction end style close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator 2 minus left parenthesis x minus 5 right parenthesis over denominator 2 times 2 end fraction end cell row blank equals cell fraction numerator 7 minus x over denominator 4 end fraction end cell end table end style  

begin mathsize 14px style 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 left parenthesis x right parenthesis end cell equals cell f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent left parenthesis x right parenthesis close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses fraction numerator 1 minus x over denominator 2 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator 1 minus x over denominator 2 end fraction end style close parentheses minus 5 over denominator 2 end fraction end cell row blank equals cell fraction numerator left parenthesis 1 minus x right parenthesis minus 5 times 2 over denominator 2 times 2 end fraction end cell row blank equals cell fraction numerator 11 minus x over denominator 4 end fraction end cell end table end style 

Jadi, fungsi invers begin mathsize 14px style f to the power of negative 1 end exponent equals fraction numerator x minus 5 over denominator 2 end fraction end stylebegin mathsize 14px style g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 1 minus x over denominator 2 end fraction end stylebegin mathsize 14px style 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 fraction numerator 7 minus x over denominator 4 end fraction end style, dan begin mathsize 14px style open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses left parenthesis x right parenthesis equals fraction numerator 11 minus x over denominator 4 end fraction end style.

0

Roboguru

Invers dari fungsi f(x)=52​x−6 adalah ....

Pembahasan Soal:

Diketahui: begin mathsize 14px style f left parenthesis x right parenthesis equals 2 over 5 x minus 6 end style

misalkan : begin mathsize 14px style y equals f open parentheses x close parentheses end style

maka invers dari begin mathsize 14px style f open parentheses x close parentheses end style:

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 cell 2 over 5 x minus 6 end cell row y equals cell 2 over 5 x minus 6 end cell row cell y plus 6 end cell equals cell 2 over 5 x end cell row cell fraction numerator 5 y plus 30 over denominator 2 end fraction end cell equals x row cell fraction numerator 5 y over denominator 2 end fraction plus 15 end cell equals x end table end style  

Oleh karena begin mathsize 14px style y equals f open parentheses x close parentheses end style maka begin mathsize 14px style x equals f to the power of negative 1 end exponent open parentheses y close parentheses end style sehingga:

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell 5 over 2 y plus 15 end cell row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell 5 over 2 y plus 15 end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell 5 over 2 x plus 15 end cell end table end style 

Jadi, jawaban yang tepat adalah E. 

 

0

Roboguru

Tentukan invers fungsi berikut: 2. g(x)=52x+1​

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x plus 1 over denominator 5 end fraction end cell row y equals cell fraction numerator 2 x plus 1 over denominator 5 end fraction end cell row cell 5 y end cell equals cell 2 x plus 1 end cell row cell 2 x end cell equals cell 5 y minus 1 end cell row x equals cell fraction numerator 5 y minus 1 over denominator 2 end fraction end cell row cell g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x minus 1 over denominator 2 end fraction end cell end table end style 

Jadi, untuk table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x plus 1 over denominator 5 end fraction end cell end table diperoleh table attributes columnalign right center left columnspacing 0px end attributes row cell g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x minus 1 over denominator 2 end fraction end cell end table.

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Roboguru

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