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Pertanyaan

Diketahui fungsi begin mathsize 14px style f colon R rightwards arrow R space dan space g colon R rightwards arrow R end style dan begin mathsize 14px style f open parentheses x close parentheses equals 4 x plus 3 end style dan begin mathsize 14px style g open parentheses x close parentheses equals x plus 2 end style. Tentukan:

begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses end style 

Pembahasan Soal:

Nilai dari begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses end style dapat dicari dengan invers fungsi komposisi sebagai 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 open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell 4 open parentheses x plus 2 close parentheses plus 3 end cell row blank equals cell 4 x plus 8 plus 3 end cell row blank equals cell 4 x plus 11 end cell row blank blank blank row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell 4 x plus 11 end cell row y equals cell 4 x plus 11 end cell row cell 4 x end cell equals cell y minus 22 end cell row x equals cell fraction numerator y minus 22 over denominator 4 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 22 over denominator 4 end fraction end cell row blank blank blank row cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis 2 right parenthesis end cell equals cell fraction numerator 2 minus 22 over denominator 4 end fraction end cell row blank equals cell negative 20 over 4 end cell row blank equals cell negative 5 end cell end table end style

Jadi, nilai dari begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses end style yaitu begin mathsize 14px style negative 5 end style.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

H. Hermawan

Mahasiswa/Alumni Universitas Lampung

Terakhir diupdate 31 Maret 2021

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Pertanyaan yang serupa

Diketahui:  dan . Tentukan invers dari !

Pembahasan Soal:

Ingat bahwa:

Jika undefined maka 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 end style 

Komposisi f(x)dang(x) adalah

fg(x)=====f(g(x))3(2x+14x5)+42x+112x15+42x+112x15+8x+42x+120x11 

Sehingga diperoleh inversnya adalah

fg1(x)==2x20x112x+20x+11

Dengan demikian, invers dari fg(x) adalah fg1(x)=2x+20x+11.

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Pembahasan Soal:

Diketahui h left parenthesis x plus 1 right parenthesis equals 3 x minus 1

Misalkan a equals x plus 1 maka a minus 1 equals x 

Sehingga,

h left parenthesis a right parenthesis equals 3 left parenthesis a minus 1 right parenthesis minus 1 h left parenthesis a right parenthesis equals 3 a minus 3 minus 1 h left parenthesis a right parenthesis equals 3 a minus 4 h left parenthesis x right parenthesis equals 3 x minus 4 

Menentukan invers dari h left parenthesis x right parenthesis 

table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis end cell equals cell 3 x minus 4 end cell row y equals cell 3 x minus 4 end cell row cell y plus 4 end cell equals cell 3 x end cell row cell fraction numerator y plus 4 over denominator 3 end fraction end cell equals x row cell fraction numerator y plus 4 over denominator 3 end fraction end cell equals cell h to the power of negative 1 end exponent left parenthesis y right parenthesis end cell row cell fraction numerator x plus 4 over denominator 3 end fraction end cell equals cell h to the power of negative 1 end exponent left parenthesis x right parenthesis end cell end table 

Menentukan left parenthesis g ring operator h to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis 

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator h to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis end cell equals cell g left parenthesis h to the power of negative 1 end exponent left parenthesis x right parenthesis right parenthesis end cell row blank equals cell g open parentheses fraction numerator x plus 4 over denominator 3 end fraction close parentheses end cell row blank equals cell fraction numerator 2 open parentheses begin display style fraction numerator x plus 4 over denominator 3 end fraction end style close parentheses over denominator open parentheses begin display style fraction numerator x plus 4 over denominator 3 end fraction end style close parentheses plus 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 x plus 8 over denominator 3 end fraction end style over denominator begin display style fraction numerator x plus 4 plus 9 over denominator 3 end fraction end style end fraction end cell row blank equals cell fraction numerator 2 x plus 8 over denominator 3 end fraction times fraction numerator 3 over denominator x plus 13 end fraction end cell row blank equals cell fraction numerator 2 x plus 8 over denominator x plus 13 end fraction end cell end table   

Menentukan left parenthesis g ring operator h to the power of negative 1 end exponent right parenthesis left parenthesis 4 right parenthesis

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator h to the power of negative 1 end exponent right parenthesis left parenthesis 4 right parenthesis end cell equals cell fraction numerator 2 left parenthesis 4 right parenthesis plus 8 over denominator 4 plus 13 end fraction end cell row blank equals cell fraction numerator 8 plus 8 over denominator 17 end fraction end cell row blank equals cell 16 over 17 end cell end table 

Jadi, nilai dari left parenthesis g ring operator h to the power of negative 1 end exponent right parenthesis left parenthesis 4 right parenthesis equals 16 over 17.

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Pembahasan Soal:

Tentukan terlebih dahulu fungsi komposisi.

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 2 x minus 1 close parentheses end cell row blank equals cell 2 x minus 1 plus 3 end cell row blank equals cell 2 x plus 2 end cell end table

Kemudian tentukan invers fungsi komposisinya.

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 2 x plus 2 end cell row y equals cell 2 x plus 2 end cell row cell 2 x end cell equals cell y minus 2 end cell row x equals cell fraction numerator y minus 2 over denominator 2 end fraction end cell row x equals cell 1 half y minus 1 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 1 half x minus 1 end cell end table

Kemudian substitusi x equals 30.

table attributes columnalign right center left columnspacing 0px 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 1 half x minus 1 end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 30 close parentheses end cell equals cell 1 half open parentheses 30 close parentheses minus 1 end cell row blank equals cell 15 minus 1 end cell row blank equals 14 end table

Maka, hasil dari open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 30 close parentheses adalah 14.

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Diketahui  dan , dengan . Jika , maka

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 squared plus 1 end cell row cell g open parentheses x close parentheses end cell equals cell a x plus 2 comma space space a not equal to 0 end cell row cell open parentheses f ring operator g to the power of negative 1 end exponent close parentheses open parentheses 1 close parentheses end cell equals 5 end table

Perhatikan fungsi g open parentheses x close parentheses equals a x plus 2 misalkan g open parentheses x close parentheses equals y maka

table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses x close parentheses end cell equals y row cell a x plus 2 end cell equals y row cell a x end cell equals cell y minus 2 end cell row x equals cell fraction numerator y minus 2 over denominator a end fraction comma space a not equal to 0 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 minus 2 over denominator a end fraction comma space a not equal to 0 end cell end table

Dengan menggunakan konsep komposisi fungsi diperoleh

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

Sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell equals 5 row cell open parentheses fraction numerator 1 minus 2 over denominator a end fraction close parentheses squared plus 1 end cell equals 5 row cell 1 over a squared end cell equals cell 5 minus 1 end cell row cell 1 over a squared end cell equals 4 row cell a squared end cell equals cell 1 fourth end cell end table

Maka

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 a squared minus 3 end cell equals cell 4 open parentheses 1 fourth close parentheses minus 3 end cell row blank equals cell 1 minus 3 end cell row blank equals cell negative 2 end cell end table

Dengan demikian nilai dari Error converting from MathML to accessible text. adalah negative 2.

Oleh karena itu, jawaban yang benar adalah B.

Roboguru

Diketahui fungsi  dan  dan . Tentukan:

Pembahasan Soal:

Nilai dari begin mathsize 14px style open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses end style dapat dicari dengan invers fungsi komposisi sebagai berikut.

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 g open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell 4 x plus 3 plus 2 end cell row blank equals cell 4 x plus 5 end cell row blank blank blank row cell left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals cell 4 x plus 5 end cell row y equals cell 4 x plus 5 end cell row cell 4 x end cell equals cell y minus 5 end cell row x equals cell fraction numerator y minus 5 over denominator 4 end fraction end cell row cell left parenthesis g ring operator f right parenthesis blank to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator x minus 5 over denominator 4 end fraction end cell row blank blank blank row cell left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis 2 right parenthesis end cell equals cell fraction numerator x minus 5 over denominator 4 end fraction end cell row blank equals cell fraction numerator 2 minus 5 over denominator 4 end fraction end cell row blank equals cell negative 3 over 4 end cell end table end style

Jadi, nilai dari begin mathsize 14px style open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses end style yaitu begin mathsize 14px style negative 3 over 4 end style.

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