Roboguru

Diketahui rumus fungsi , , dan . Tentukan rumus fungsi   dan nilai  agar !

Pertanyaan

Diketahui rumus fungsi f open parentheses x close parentheses equals fraction numerator x minus 5 over denominator 2 end fractiong to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x over denominator x minus 4 end fraction, dan h open parentheses x close parentheses equals x over 4. Tentukan rumus fungsi g open parentheses x close parentheses  dan nilai k agar open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses k close parentheses equals 3!

Pembahasan Soal:

Diketahui fungsi f open parentheses x close parentheses equals fraction numerator x minus 5 over denominator 2 end fractiong to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x over denominator x minus 4 end fraction, dan h open parentheses x close parentheses equals x over 4. Rumus fungsi g open parentheses x close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell g to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator x over denominator x minus 4 end fraction end cell row y equals cell fraction numerator x over denominator x minus 4 end fraction end cell row cell y open parentheses x minus 4 close parentheses end cell equals x row cell x y minus 4 y end cell equals x row cell x y minus x end cell equals cell 4 y end cell row cell x open parentheses y minus 1 close parentheses end cell equals cell 4 y end cell row x equals cell fraction numerator 4 y over denominator y minus 1 end fraction end cell row cell g open parentheses y close parentheses end cell equals cell fraction numerator 4 y over denominator y minus 1 end fraction end cell row cell g open parentheses x close parentheses end cell equals cell fraction numerator 4 x over denominator x minus 1 end fraction end cell end table

Kemudian, diketahui open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses k close parentheses equals 3. Untuk menentukan nilai k, harus ditentukan dulu rumus fungsi open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses. Rumus fungsi open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses dapat ditentukan dengan menggunakan rumus fungsi open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses. Rumus fungsi open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses sebagai berikut.

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 open parentheses h ring operator g close parentheses open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell h open parentheses g open parentheses f open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell h open parentheses g open parentheses fraction numerator x minus 5 over denominator 2 end fraction close parentheses close parentheses end cell row blank equals cell h open parentheses fraction numerator 4 open parentheses begin display style fraction numerator x minus 5 over denominator 2 end fraction end style close parentheses over denominator begin display style fraction numerator x minus 5 over denominator 2 end fraction end style minus 1 end fraction close parentheses end cell row blank equals cell h open parentheses fraction numerator 2 open parentheses x minus 5 close parentheses over denominator begin display style fraction numerator x minus 5 minus 2 over denominator 2 end fraction end style end fraction close parentheses end cell row blank equals cell h open parentheses 2 open parentheses x minus 5 close parentheses times fraction numerator 2 over denominator open parentheses x minus 7 close parentheses end fraction close parentheses end cell row blank equals cell h open parentheses fraction numerator 4 open parentheses x minus 5 close parentheses over denominator x minus 7 end fraction close parentheses end cell row blank equals cell fraction numerator begin display style fraction numerator 4 open parentheses x minus 5 close parentheses over denominator x minus 7 end fraction end style over denominator 4 end fraction end cell row blank equals cell fraction numerator 4 open parentheses x minus 5 close parentheses over denominator x minus 7 end fraction times 1 fourth end cell row blank equals cell fraction numerator x minus 5 over denominator x minus 7 end fraction end cell end table

Lalu, open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses sebagai berikut.

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 minus 5 over denominator x minus 7 end fraction end cell row y equals cell fraction numerator x minus 5 over denominator x minus 7 end fraction end cell row cell y open parentheses x minus 7 close parentheses end cell equals cell x minus 5 end cell row cell x y minus 7 y end cell equals cell x minus 5 end cell row cell x y minus x end cell equals cell negative 5 plus 7 y end cell row cell x open parentheses y minus 1 close parentheses end cell equals cell 7 y minus 5 end cell row x equals cell fraction numerator 7 y minus 5 over denominator y minus 1 end fraction 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 y close parentheses end cell equals cell fraction numerator 7 y minus 5 over denominator y minus 1 end fraction 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 fraction numerator 7 x minus 5 over denominator x minus 1 end fraction end cell row blank blank blank end table

Nilai k yang memenuhi sebagai berikut.

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 k close parentheses end cell equals 3 row cell fraction numerator 7 k minus 5 over denominator k minus 1 end fraction end cell equals 3 row cell 7 k minus 5 end cell equals cell 3 open parentheses k minus 1 close parentheses end cell row cell 7 k minus 5 end cell equals cell 3 k minus 3 end cell row cell 7 k minus 3 k end cell equals cell negative 3 plus 5 end cell row cell 4 k end cell equals 2 row k equals cell 2 over 4 end cell row blank equals cell 1 half end cell row blank blank blank row blank blank blank end table

Jadi, g open parentheses x close parentheses equals fraction numerator 4 x over denominator x minus 1 end fraction dan nilai k yang memenuhi open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses k close parentheses equals 3 yaitu k equals 1 half.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. Hajrianti

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 02 Mei 2021

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Pertanyaan yang serupa

Diketahui fungsi sebagai berikut:   Tentukan: e.

Pembahasan Soal:

Gunakan konsep komposisi tiga fungsi.

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

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 2 x plus 1 end cell row cell h open parentheses x close parentheses end cell equals cell x minus 4 end cell row cell g open parentheses x close parentheses end cell equals cell x squared minus 3 x plus 2 end cell row cell m open parentheses x close parentheses end cell equals cell fraction numerator x plus 4 over denominator 2 x minus 5 end fraction end cell end table end style

Akan ditentukan begin mathsize 14px style open parentheses m ring operator f ring operator g close parentheses open parentheses x close parentheses end style, perhatikan perhitungan berikut:

Terlebih dahulu tentukan begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style,

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

Diperoleh begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses equals 2 x squared minus 6 x plus 5 end style, sehingga begin mathsize 14px style open parentheses m ring operator f ring operator g close parentheses open parentheses x close parentheses end style dapat dihitung sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses m ring operator f ring operator g close parentheses open parentheses x close parentheses end cell equals cell m ring operator open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell row blank equals cell m open parentheses open parentheses f ring operator g close parentheses open parentheses x close parentheses close parentheses end cell row blank equals cell m open parentheses 2 x squared minus 6 x plus 5 close parentheses end cell row blank equals cell fraction numerator open parentheses 2 x squared minus 6 x plus 5 close parentheses plus 4 over denominator 2 open parentheses 2 x squared minus 6 x plus 5 close parentheses minus 5 end fraction end cell row blank equals cell fraction numerator 2 x squared minus 6 x plus 9 over denominator 4 x squared minus 12 x plus 10 minus 5 end fraction end cell row cell open parentheses m ring operator f ring operator g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator 2 x squared minus 6 x plus 9 over denominator 4 x squared minus 12 x plus 5 end fraction end cell end table end style 

Jadi, diperoleh begin mathsize 14px style open parentheses m ring operator f ring operator g close parentheses open parentheses x close parentheses equals fraction numerator 2 x squared minus 6 x plus 9 over denominator 4 x squared minus 12 x plus 5 end fraction end style.

Roboguru

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.

Roboguru

Diketahui , , dan . Jika , nilai dari

Pembahasan Soal:

Fungsi komposisi tiga fungsi tersebut adalah sebagai 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 open parentheses x close parentheses end cell equals cell open parentheses f ring operator open parentheses g ring operator h close parentheses close parentheses open parentheses x close parentheses end cell row blank 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 4 minus 3 open parentheses 2 x plus 3 close parentheses close parentheses end cell row blank equals cell f open parentheses negative 6 x minus 5 close parentheses end cell row blank equals cell open parentheses negative 6 x minus 5 close parentheses plus 2 end cell row blank equals cell negative 6 x minus 3 end cell end table end style

Invers dari fungsi komposisi tersebut adalah sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative 6 x minus 3 end cell row cell 6 x end cell equals cell negative y minus 3 end cell row x equals cell fraction numerator negative y minus 3 over denominator 6 end fraction end cell row cell open parentheses f ring operator g ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative x minus 3 over denominator 6 end fraction end cell end table end style

Jika begin mathsize 14px style open parentheses f ring operator g ring operator h close parentheses to the power of negative 1 end exponent open parentheses text a end text close parentheses equals 1 end style, maka nilai begin mathsize 14px style text a end text end style dapat ditentukan sebagai 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 to the power of negative 1 end exponent open parentheses text a end text close parentheses end cell equals 1 row cell fraction numerator negative text a end text minus 3 over denominator 6 end fraction end cell equals 1 row cell negative text a end text minus 3 end cell equals 6 row cell negative text a end text end cell equals 9 row cell text a end text end cell equals cell negative 9 end cell end table end style

Oleh karena itu, jawaban yang tepat adalah A.

 

Roboguru

Jika , , dan , fungsi komposisi

Pembahasan Soal:

Fungsi size 14px g size 14px ring operator size 14px h size 14px ring operator size 14px f artinya fungsi size 14px f dikerjakan terlebih dahulu, lalu dilanjutkan dengan fungsi begin mathsize 14px style h end style, dan terakhir fungsi begin mathsize 14px style g end style.

Karena begin mathsize 14px style f open parentheses x close parentheses equals 2 x minus 1 end stylebegin mathsize 14px style g open parentheses x close parentheses equals x squared plus 2 end style, dan begin mathsize 14px style h open parentheses x close parentheses equals fraction numerator 1 over denominator x minus 1 end fraction end style, maka

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 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 2 x minus 1 close parentheses close parentheses end cell row blank equals cell g open parentheses fraction numerator 1 over denominator open parentheses 2 x minus 1 close parentheses minus 1 end fraction close parentheses end cell row blank equals cell g open parentheses fraction numerator 1 over denominator 2 x minus 2 end fraction close parentheses end cell row blank equals cell open parentheses fraction numerator 1 over denominator 2 x minus 2 end fraction close parentheses squared plus 2 end cell row blank equals cell fraction numerator 1 over denominator 4 x squared minus 8 x plus 4 end fraction plus 2 end cell row blank equals cell fraction numerator 1 plus 2 open parentheses 4 x squared minus 8 x plus 4 close parentheses over denominator 4 x squared minus 8 x plus 4 end fraction end cell row blank equals cell fraction numerator 1 plus 8 x squared minus 16 x plus 8 over denominator 4 x squared minus 8 x plus 4 end fraction end cell row blank equals cell fraction numerator 8 x squared minus 16 x plus 9 over denominator 4 x squared minus 8 x plus 4 end fraction end cell end table end style

Jadi, jawaban yang tepat adalah A.

Roboguru

Diketahui  dan . Tentukan rumus fungsi berikut a.

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 h ring operator f ring operator g close parentheses open parentheses x close parentheses end style = ?

Jawab:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses h ring operator f ring operator g close parentheses open parentheses x close parentheses end cell equals cell h open parentheses f open parentheses g open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell h open parentheses f open parentheses 3 x minus 1 close parentheses close parentheses end cell row blank equals cell h open parentheses 3 x minus 1 plus 2 close parentheses end cell row blank equals cell open parentheses 3 x plus 1 close parentheses squared minus 1 end cell row blank equals cell 9 x squared plus 6 x plus 1 minus 1 end cell row blank equals cell 9 x squared plus 6 x end cell end table 

Jadi, rumus fungsi begin mathsize 14px style open parentheses h ring operator f ring operator g close parentheses open parentheses x close parentheses end style adalah 9 x squared plus 6 x.

 

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