Roboguru

Diketahui  dan . Tentukan rumus fungsi berikut b.

Pertanyaan

Diketahui begin mathsize 14px style f left parenthesis x right parenthesis equals x plus 2 comma space g left parenthesis x right parenthesis equals 3 x minus 1 end style dan begin mathsize 14px style h left parenthesis x right parenthesis equals x squared minus 1 end style. Tentukan rumus fungsi berikut

b. begin mathsize 14px style open parentheses g ring operator h ring operator f close parentheses open parentheses x close parentheses end style  

Pembahasan Soal:

Diketahui:

begin mathsize 14px style f left parenthesis x right parenthesis equals x plus 2 comma space g left parenthesis x right parenthesis equals 3 x minus 1 end stylebegin mathsize 14px style h left parenthesis x right parenthesis equals x squared minus 1 end style

Ditanya:

begin mathsize 14px style open parentheses g ring operator h ring operator f close parentheses open parentheses x close parentheses end style

Jawab:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator h ring operator f close parentheses open parentheses x close parentheses end cell equals cell g open parentheses h open parentheses f open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell g open parentheses h open parentheses x plus 2 close parentheses close parentheses end cell row blank equals cell g open parentheses open parentheses x plus 2 close parentheses squared minus 1 close parentheses end cell row blank equals cell g open parentheses x squared plus 4 x plus 4 minus 1 close parentheses end cell row blank equals cell 3 open parentheses x squared plus 4 x plus 3 close parentheses minus 1 end cell row blank equals cell 3 x squared plus 12 x plus 9 minus 1 end cell row blank equals cell 3 x squared plus 12 x plus 8 end cell end table 

Jadi, rumus fungsi begin mathsize 14px style open parentheses g ring operator h ring operator f close parentheses open parentheses x close parentheses end style adalah 3 x squared plus 12 x plus 8

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Mustikowati

Mahasiswa/Alumni Universitas Negeri Jakarta

Terakhir diupdate 03 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Diketahui fungsi , , dan . Tentukan : c.  dan

Pembahasan Soal:

Terlebih dahulu dicari fungsi komposisi begin mathsize 14px style open parentheses g ring operator h close parentheses left parenthesis x right parenthesis end style, yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator h close parentheses left parenthesis x right parenthesis end cell equals cell g left parenthesis h left parenthesis x right parenthesis right parenthesis end cell row blank equals cell g open parentheses x squared minus 1 close parentheses end cell row blank equals cell 3 open parentheses x squared minus 1 close parentheses minus 1 end cell row blank equals cell 3 x squared minus 3 minus 1 end cell row blank equals cell 3 x squared minus 4 end cell end table end style 

Selanjutnya fungsi komposisi begin mathsize 14px style open parentheses g ring operator h close parentheses left parenthesis x right parenthesis end style disubstitusikan ke dalam fungsi size 14px f size 14px left parenthesis size 14px x size 14px right parenthesis seperti berikut,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g ring operator h close parentheses left parenthesis x right parenthesis end cell equals cell f left parenthesis g left parenthesis h left parenthesis x right parenthesis right parenthesis right parenthesis end cell row blank equals cell f open parentheses 3 x squared minus 4 close parentheses end cell row blank equals cell open parentheses 3 x squared minus 4 close parentheses plus 2 end cell row blank equals cell 3 x squared minus 2 end cell end table end style 

Dengan demikian komposisi dari ketiga fungsi begin mathsize 14px style open parentheses f ring operator g ring operator h close parentheses left parenthesis x right parenthesis end style adalah begin mathsize 14px style 3 x squared minus 2 end style

Nilai begin mathsize 14px style x equals negative 2 end style disubstitusikan ke dalam fungsi komposisi sehingga didapatkan,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g ring operator h close parentheses left parenthesis negative 2 right parenthesis end cell equals cell 3 times left parenthesis negative 2 right parenthesis squared minus 2 end cell row blank equals cell 12 minus 2 end cell row blank equals 10 end table end style  

Jadi, nilai dari undefined adalah begin mathsize 14px style 10 end style.

0

Roboguru

1. Diket : Tentukan  b)

Pembahasan Soal:

Diketahui :

straight f open parentheses straight x close parentheses equals straight x plus 3 straight g open parentheses straight x close parentheses equals 2 straight x minus 1 straight h open parentheses straight x close parentheses equals straight x squared

Ditanya :

open parentheses straight h ring operator straight g ring operator straight f close parentheses open parentheses 3 close parentheses

Penyelesaian 

open parentheses straight h ring operator straight g ring operator straight f close parentheses open parentheses 3 close parentheses

Akan dicari fungsi komposisi dari open parentheses g ring operator f close parentheses open parentheses x close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses straight g ring operator straight f close parentheses open parentheses straight x close parentheses end cell equals cell straight g open parentheses straight f open parentheses straight x close parentheses close parentheses end cell row blank equals cell 2 open parentheses straight x plus 3 close parentheses minus 1 end cell row blank equals cell 2 straight x plus 6 minus 1 end cell row blank equals cell 2 straight x plus 5 end cell end table

selanjutnya akan dicari komposisi fungsi dari open parentheses straight h ring operator straight g ring operator straight f close parentheses open parentheses x close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses straight h ring operator straight g ring operator straight f close parentheses open parentheses straight x close parentheses end cell equals cell straight h open parentheses straight g ring operator straight f open parentheses straight x close parentheses close parentheses end cell row blank equals cell straight h open parentheses 2 straight x plus 5 close parentheses end cell row blank equals cell open parentheses 2 straight x plus 5 close parentheses squared end cell row blank equals cell 4 straight x squared plus 20 straight x plus 25 end cell end table

Akan dicari nilai dari open parentheses straight h ring operator straight g ring operator straight f close parentheses open parentheses 3 close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses straight h ring operator straight g ring operator straight f close parentheses open parentheses 3 close parentheses end cell equals cell 4 open parentheses 3 close parentheses squared plus 20 open parentheses 3 close parentheses plus 25 end cell row blank equals cell 4 open parentheses 9 close parentheses plus 60 plus 25 end cell row blank equals cell 36 plus 60 plus 25 end cell row blank equals 121 end table

Jadi, nilai dari open parentheses straight h ring operator straight g ring operator straight f close parentheses open parentheses 3 close parentheses equals 121.

0

Roboguru

Diketahui fungsi , , dan  dengan , , dan . Tentukan: b.

Pembahasan Soal:

Komposisi dari tiga fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals x plus 2 end stylebegin mathsize 14px style g left parenthesis x right parenthesis equals 3 x minus 1 end style, dan begin mathsize 14px style h left parenthesis x right parenthesis equals x squared minus 1 end style dapat dikerjakan dari fungsi paling kanan terlebih dahulu.

Pertama, dicari fungsi komposisi begin mathsize 14px style open parentheses h ring operator f close parentheses left parenthesis x right parenthesis end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses h ring operator f close parentheses left parenthesis x right parenthesis end cell equals cell h left parenthesis f left parenthesis x right parenthesis right parenthesis end cell row blank equals cell h left parenthesis x plus 2 right parenthesis end cell row blank equals cell left parenthesis x plus 2 right parenthesis squared minus 1 end cell row blank equals cell x squared plus 4 x plus 4 minus 1 end cell row blank equals cell x squared plus 4 x plus 3 end cell end table end style 

Selanjutnya, fungsi begin mathsize 14px style open parentheses h ring operator f close parentheses left parenthesis x right parenthesis end style dikomposisikan dengan fungsi begin mathsize 14px style g left parenthesis x right parenthesis end style menjadi,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator h ring operator f close parentheses left parenthesis x right parenthesis end cell equals cell g left parenthesis h left parenthesis f left parenthesis x right parenthesis right parenthesis right parenthesis end cell row blank equals cell g open parentheses x squared plus 4 x plus 3 close parentheses end cell row blank equals cell 3 open parentheses x squared plus 4 x plus 3 close parentheses minus 1 end cell row blank equals cell 3 x squared plus 12 x plus 9 minus 1 end cell row blank equals cell 3 x squared plus 12 x plus 8 end cell end table end style 

Jadi, hasil dari fungsi komposisi begin mathsize 14px style open parentheses g ring operator h ring operator f close parentheses left parenthesis x right parenthesis end style adalah Error converting from MathML to accessible text..

0

Roboguru

Diketahui , , dan  berturut-turut menyatakan invers fungsi , , dan . Jika  dan , nilai  adalah ...

Pembahasan Soal:

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 ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals 2 x minus 4 end style dengan menggunakan konsep invers fungsi komposisi maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent end cell equals cell 2 x minus 4 end cell row cell open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell 2 x minus 4 end cell end table end style 

Misalkan begin mathsize 14px style open parentheses table attributes columnalign right center left columnspacing 0px end attributes row blank blank h end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank ring operator end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank ring operator end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank f end table close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals y end style maka begin mathsize 14px style x equals open parentheses table attributes columnalign right center left columnspacing 0px end attributes row blank blank h end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank ring operator end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank ring operator end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank f end table close parentheses open parentheses y close parentheses end style sehingga:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell 2 x minus 4 end cell row y equals cell 2 x minus 4 end cell row cell y plus 4 end cell equals cell 2 x end cell row x equals cell fraction numerator y plus 4 over denominator 2 end fraction end cell row cell open parentheses h ring operator g ring operator f close parentheses open parentheses y close parentheses end cell equals cell fraction numerator y plus 4 over denominator 2 end fraction end cell row cell open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses end cell equals cell fraction numerator x plus 4 over denominator 2 end fraction end cell end table end style 

begin mathsize 14px style open parentheses h ring operator g close parentheses open parentheses x close parentheses equals fraction numerator x minus 3 over denominator 2 x plus 1 end fraction comma space x not equal to 1 half end style maka berdasarkan konsep komposisi fungsi:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses end cell equals cell fraction numerator x plus 4 over denominator 2 end fraction end cell row cell open parentheses h ring operator g close parentheses open parentheses f open parentheses x close parentheses close parentheses end cell equals cell fraction numerator x plus 4 over denominator 2 end fraction end cell row cell fraction numerator f open parentheses x close parentheses minus 3 over denominator 2 f open parentheses x close parentheses plus 1 end fraction end cell equals cell fraction numerator x plus 4 over denominator 2 end fraction end cell row cell 2 open parentheses f open parentheses x close parentheses minus 3 close parentheses end cell equals cell open parentheses x plus 4 close parentheses open parentheses 2 f open parentheses x close parentheses plus 1 close parentheses end cell row cell 2 f open parentheses x close parentheses minus 6 end cell equals cell 2 x f open parentheses x close parentheses plus 8 f open parentheses x close parentheses plus x plus 4 end cell row cell 2 f open parentheses x close parentheses minus 8 f open parentheses x close parentheses minus 2 x f open parentheses x close parentheses end cell equals cell x plus 4 plus 6 end cell row cell negative 6 f open parentheses x close parentheses minus 2 x f open parentheses x close parentheses end cell equals cell x plus 10 end cell row cell f open parentheses x close parentheses open parentheses negative 6 minus 2 x close parentheses end cell equals cell x plus 10 end cell row cell f open parentheses x close parentheses end cell equals cell fraction numerator x plus 10 over denominator negative 6 minus 2 x end fraction end cell row cell f open parentheses 8 close parentheses end cell equals cell fraction numerator 8 plus 10 over denominator negative 6 minus 2 open parentheses 8 close parentheses end fraction end cell row blank equals cell fraction numerator 18 over denominator negative 22 end fraction end cell row blank equals cell negative 9 over 11 end cell end table end style 

Oleh karena itu, jawaban yang benar adalah C.

0

Roboguru

tentukan (f∘(g∘h))(x)=....

Pembahasan Soal:

Diketahui

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px 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 2 x minus 2 end cell row cell h open parentheses x close parentheses end cell equals cell 3 x plus 2 end cell end table end style

Akan ditentukan (f(gh))(x)

Perhatikan perhitungan berikut

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses f ring operator open parentheses g ring operator h close parentheses close parentheses open parentheses x close parentheses end cell equals cell f open parentheses g open parentheses h open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell f open parentheses g open parentheses 3 x plus 2 close parentheses close parentheses end cell row blank equals cell f open parentheses 2 open parentheses 3 x plus 2 close parentheses minus 2 close parentheses end cell row blank equals cell f open parentheses 6 x plus 4 minus 2 close parentheses end cell row blank equals cell f open parentheses 6 x plus 2 close parentheses end cell row blank equals cell open parentheses 6 x plus 2 close parentheses squared plus 1 end cell row blank equals cell 36 x squared plus 24 x plus 4 plus 1 end cell row blank equals cell 36 x squared plus 24 x plus 5 end cell end table end style 

Oleh karena itu, jawaban yang tepat adalah E

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