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Diberikan. Invers dari  adalah

Pertanyaan

Diberikanf open parentheses x close parentheses equals 5 x minus 4. Invers dari f open parentheses x close parentheses adalah f to the power of negative 1 end exponent open parentheses x close parentheses equals...

Pembahasan Soal:

Dengan menggunakan konsep invers fungsi maka

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals y row cell 5 x minus 4 end cell equals y row cell 5 x end cell equals cell y plus 4 end cell row x equals cell fraction numerator y plus 4 over denominator 5 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 4 over denominator 5 end fraction end cell end table

Dengan demikian inver dari f open parentheses x close parentheses adalah f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x plus 4 over denominator 5 end fraction.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Nur

Mahasiswa/Alumni Institut Teknologi Sepuluh Nopember

Terakhir diupdate 02 Mei 2021

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Pertanyaan yang serupa

Diketahui fungsi . Tentukan: a. Invers fungsi  b. Nilai dari

Pembahasan Soal:

Diketahui fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals 4 x plus 2 space dan space left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis equals 8 x plus 3 end style.

Menentukan begin mathsize 14px style g 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 g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals cell 8 x plus 3 end cell row cell g left parenthesis f left parenthesis x right parenthesis right parenthesis end cell equals cell 8 x plus 3 end cell row cell g left parenthesis 4 x plus 2 right parenthesis end cell equals cell 8 x plus 3 end cell row blank blank blank end table end style 

Misalkan begin mathsize 14px style a equals 4 x plus 2 space end stylemaka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a minus 2 end cell equals cell 4 x end cell row cell fraction numerator a minus 2 over denominator 4 end fraction end cell equals x end table end style 

Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis a right parenthesis end cell equals cell 8 left parenthesis fraction numerator a minus 2 over denominator 4 end fraction right parenthesis plus 3 end cell row cell g left parenthesis a right parenthesis end cell equals cell 2 left parenthesis a minus 2 right parenthesis plus 3 end cell row cell g left parenthesis a right parenthesis end cell equals cell 2 a minus 4 plus 3 end cell row cell g left parenthesis a right parenthesis end cell equals cell 2 a minus 1 end cell row cell g left parenthesis x right parenthesis end cell equals cell 2 x minus 1 end cell end table end style 

Menentukan invers fungsi begin mathsize 14px style g end style

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 2 x minus 1 end cell row y equals cell 2 x minus 1 end cell row cell y plus 1 end cell equals cell 2 x end cell row cell fraction numerator y plus 1 over denominator 2 end fraction end cell equals x row cell fraction numerator y plus 1 over denominator 2 end fraction end cell equals cell g to the power of negative 1 end exponent left parenthesis y right parenthesis end cell row cell fraction numerator x plus 1 over denominator 2 end fraction end cell equals cell g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell end table end style 

a. Jadi, begin mathsize 14px style g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator x plus 1 over denominator 2 end fraction end style.

  begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g to the power of negative 1 end exponent left parenthesis 7 right parenthesis end cell equals cell fraction numerator 7 plus 1 over denominator 2 end fraction end cell row blank equals cell 8 over 2 end cell row blank equals 4 end table end style 

b. Jadi, nilai dari begin mathsize 14px style g to the power of negative 1 end exponent left parenthesis 7 right parenthesis equals 4 end style.

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Roboguru

Tentukan  dari .

Pembahasan Soal:

Dengan menggunakan konsep invers fungsi, maka didapatkan:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell fraction numerator x plus 1 over denominator x minus 2 end fraction end cell row y equals cell fraction numerator x plus 1 over denominator x minus 2 end fraction end cell row cell y x minus 2 y end cell equals cell x plus 1 end cell row cell y x minus x end cell equals cell 2 y plus 1 end cell row cell x left parenthesis y minus 1 right parenthesis end cell equals cell 2 y plus 1 end cell row x equals cell fraction numerator 2 y plus 1 over denominator y minus 1 end fraction end cell row blank blank blank 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 1 over denominator x minus 1 end fraction end cell end table end style

Jadi, jawabannya yaitu begin mathsize 14px style 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 2 x plus 1 over denominator x minus 1 end fraction end cell end table end style.

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Roboguru

Jika  dan  maka

Pembahasan Soal:

Diketahui open parentheses f ring operator g close parentheses open parentheses x close parentheses equals 10 x squared minus 8 x minus 3 dan g open parentheses x close parentheses equals 2 x plus 4 maka:

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 f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell row cell 10 x squared minus 8 x minus 3 end cell equals cell f left parenthesis 2 x plus 4 right parenthesis end cell row cell misal space 2 x plus 4 end cell equals cell y comma space maka space x equals fraction numerator y minus 4 over denominator 2 end fraction end cell row cell 10 open parentheses fraction numerator y minus 4 over denominator 2 end fraction close parentheses squared minus 8 open parentheses fraction numerator y minus 4 over denominator 2 end fraction close parentheses minus 3 end cell equals cell f open parentheses 2 open parentheses fraction numerator y minus 4 over denominator 2 end fraction close parentheses plus 4 close parentheses end cell row cell 10 open parentheses fraction numerator y squared minus 8 y plus 16 over denominator 4 end fraction close parentheses minus 4 y plus 16 minus 3 end cell equals cell f left parenthesis y right parenthesis end cell row cell fraction numerator 10 y squared over denominator 4 end fraction minus fraction numerator 80 y over denominator 4 end fraction plus 160 over 4 minus 4 y plus 13 end cell equals cell f left parenthesis y right parenthesis end cell row cell 5 over 2 y squared minus 20 y plus 40 minus 4 y plus 13 end cell equals cell f left parenthesis y right parenthesis end cell row cell 5 over 2 y squared minus 24 y plus 53 end cell equals cell f left parenthesis y right parenthesis end cell row cell f left parenthesis x right parenthesis end cell equals cell 5 over 2 x squared minus 24 x plus 53 end cell end table 

Sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell 5 over 2 x squared minus 24 x plus 53 end cell row y equals cell 5 over 2 x squared minus 24 x plus 53 end cell row cell 2 y end cell equals cell 5 x squared minus 48 x plus 106 end cell end table

Error converting from MathML to accessible text.

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction end cell row blank equals cell fraction numerator negative left parenthesis negative 48 right parenthesis plus-or-minus square root of left parenthesis negative 48 right parenthesis squared minus 4 left parenthesis 5 right parenthesis left parenthesis 106 minus 2 y right parenthesis end root over denominator 2 left parenthesis 5 right parenthesis end fraction end cell row blank equals cell fraction numerator 48 plus-or-minus square root of 2.304 minus 2.120 plus 40 y end root over denominator 10 end fraction end cell row blank equals cell fraction numerator 48 plus-or-minus square root of 40 y plus 184 end root over denominator 10 end fraction end cell row blank equals cell fraction numerator 2 left parenthesis 24 plus-or-minus square root of 10 y plus 46 end root right parenthesis over denominator 10 end fraction end cell row blank equals cell fraction numerator left parenthesis 24 plus-or-minus square root of 10 y plus 46 end root right parenthesis over denominator 5 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 left parenthesis 24 plus-or-minus square root of 10 x plus 46 end root right parenthesis over denominator 5 end fraction end cell end table 

Jadi jawabannya tidak ada pada opsi, jawaban yang benar 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 left parenthesis 24 plus-or-minus square root of 10 x plus 46 end root right parenthesis over denominator 5 end fraction end cell end table 

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Roboguru

Tentukan rumus fungsi invers tiap fungsi berikut. a.

Pembahasan Soal:

begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 2 over denominator 2 x minus 4 end fraction comma space x not equal to 2 end style 

Misalkan begin mathsize 14px style f open parentheses x close parentheses equals y end style, maka:

table attributes columnalign right center left columnspacing 2px end attributes row cell fraction numerator size 14px 2 over denominator size 14px 2 size 14px x size 14px minus size 14px 4 end fraction end cell size 14px equals size 14px y row size 14px 2 size 14px equals cell size 14px y begin mathsize 14px style left parenthesis 2 x minus 4 right parenthesis end style end cell row cell size 14px 2 over size 14px y end cell size 14px equals cell size 14px 2 size 14px x size 14px minus size 14px 4 end cell row cell size 14px 2 over size 14px y size 14px plus size 14px 4 end cell size 14px equals cell size 14px 2 size 14px x end cell row x size 14px equals cell size 14px 1 over size 14px y size 14px plus size 14px 2 size 14px space end cell row cell size 14px f to the power of size 14px minus size 14px 1 end exponent begin mathsize 14px style left parenthesis y right parenthesis end style end cell size 14px equals cell size 14px 1 over size 14px y size 14px plus size 14px 2 size 14px semicolon size 14px space size 14px y size 14px not equal to size 14px 0 end cell end table

Jadi, rumus fungsi invers begin mathsize 14px style f end style adalah begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses equals 1 over x plus 2 space comma space x not equal to 0 end style.

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Roboguru

Diketahui  jika  adalah invers dari , maka

Pembahasan Soal:

Misalkan begin mathsize 14px style y equals f left parenthesis x right parenthesis end style, maka didapat

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

Jadi, invers dari begin mathsize 14px style bold italic f end style adalah begin mathsize 14px style bold italic f to the power of bold minus bold 1 end exponent bold equals fraction numerator bold 2 bold x bold plus bold 2 over denominator bold 3 end fraction end styleundefined 

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Roboguru

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