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Pertanyaan

Fungsi f dan g ditentukan oleh rumus f left parenthesis x right parenthesis equals x squared plus 1 dan g left parenthesis x right parenthesis equals fraction numerator 1 over denominator 2 x minus 1 end fraction. Tentukan open parentheses f plus g close parentheses left parenthesis x right parenthesis dan open parentheses f plus g close parentheses left parenthesis 2 right parenthesis.

Pembahasan Soal:

Dengan menggunakan sifat fungsi didapatkan langkah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f plus g close parentheses left parenthesis x right parenthesis end cell equals cell f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end cell row blank equals cell x squared plus 1 plus fraction numerator 1 over denominator 2 x minus 1 end fraction end cell row blank equals cell fraction numerator open parentheses x squared plus 1 close parentheses left parenthesis 2 x minus 1 right parenthesis plus 1 over denominator 2 x minus 1 end fraction end cell row blank equals cell fraction numerator 2 x cubed minus x squared plus 2 x minus 1 plus 1 over denominator 2 x minus 1 end fraction end cell row blank equals cell fraction numerator 2 x cubed minus x squared plus 2 x over denominator 2 x minus 1 end fraction end cell row blank equals cell fraction numerator x left parenthesis 2 x squared minus x plus 2 right parenthesis over denominator 2 x minus 1 end fraction end cell end table

Maka open parentheses f plus g close parentheses left parenthesis x right parenthesis adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x left parenthesis 2 x squared minus x plus 2 right parenthesis over denominator 2 x minus 1 end fraction end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f plus g right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator x left parenthesis 2 x squared minus x plus 2 right parenthesis over denominator 2 x minus 1 end fraction end cell row cell left parenthesis f plus g right parenthesis left parenthesis 2 right parenthesis end cell equals cell fraction numerator 2 left parenthesis 2 left parenthesis 2 right parenthesis squared minus left parenthesis 2 right parenthesis plus 2 right parenthesis over denominator 2 left parenthesis 2 right parenthesis minus 1 end fraction end cell row blank equals cell 16 over 3 end cell end table

Maka open parentheses f plus g close parentheses left parenthesis 2 right parenthesis adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 16 over 3 end cell end table

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Nadidah

Mahasiswa/Alumni Universitas Nasional

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 fungsi-fungsi berikut dan tentukan pula daerah asalnya. a.

Pembahasan Soal:

begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses equals square root of x squared minus 4 end root plus square root of x minus 2 end root end style

Agar fungsi terdefinisi maka harus terpenuhi syarat berikut

a. begin mathsize 14px style square root of x squared minus 4 end root greater or equal than 0 end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus 4 end cell greater or equal than 0 row cell open parentheses x plus 2 close parentheses open parentheses x minus 2 close parentheses end cell greater or equal than 0 end table end style

Sehingga nilai begin mathsize 14px style x end style yang memenuhi begin mathsize 14px style x less or equal than negative 2 space atau space x greater or equal than 2 end style

b. begin mathsize 14px style square root of x minus 2 end root greater or equal than 0 end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 2 end cell greater or equal than 0 row x greater or equal than 2 end table end style

Sehingga nilai undefined yang memenuhi begin mathsize 14px style x greater or equal than 2 end style.

Selanjutnya diperoleh irisan dari poin a dan b adalah undefined.

Dengan demikian daerah asal dari begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses end style adalah begin mathsize 14px style open curly brackets x left enclose x greater or equal than 2 comma space x element of R end enclose close curly brackets end style.

Roboguru

Jika diketahui  dan , daerah asal (domain) fungsi  adalah ...

Pembahasan Soal:

Jika diketahui f left parenthesis x right parenthesis equals fraction numerator 2 over denominator x squared minus 9 end fraction dan g left parenthesis x right parenthesis equals 2 x, daerah asal (domain) fungsi open parentheses f plus g close parentheses left parenthesis x right parenthesis adalah ...

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f plus g close parentheses left parenthesis x right parenthesis end cell equals cell f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end cell row blank equals cell fraction numerator 2 over denominator x squared minus 9 end fraction plus 2 x end cell row blank equals cell fraction numerator 2 plus 2 x left parenthesis x squared minus 9 right parenthesis over denominator x squared minus 9 end fraction end cell row blank equals cell fraction numerator 2 plus 2 x cubed minus 18 x over denominator x squared minus 9 end fraction end cell row blank equals cell fraction numerator 2 x cubed minus 18 x plus 2 over denominator left parenthesis x plus 3 right parenthesis left parenthesis x minus 3 right parenthesis end fraction end cell end table 

Agar terdefinisi, maka daerah asal fungsi tersebut mengharuskan penyebutnot equal to 0, sehingga:

left parenthesis x plus 3 right parenthesis left parenthesis x minus 3 right parenthesis not equal to 0 x not equal to negative 3 space atau space x not equal to 3  

Maka daerah asalnya adalah open curly brackets x left enclose x not equal to negative 3 space atau space x not equal to 3 comma space x element of R end enclose close curly brackets.

Jadi, jawaban yang tepat adalah D

Roboguru

Jika diketahui  dan , daerah asal (domain) fungsi  adalah ....

Pembahasan Soal:

begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 2 over denominator x squared minus 9 end fraction end style fungsi begin mathsize 14px style f open parentheses x close parentheses end style terdefinisi (mempunyai nilai) jika penyebutnya tidak sama dengan 0 begin mathsize 14px style open parentheses x squared minus 9 not equal to 0 space atau space x not equal to plus-or-minus 3 close parentheses end style. Dengan demikian, daerah asal begin mathsize 14px style f open parentheses x close parentheses end style adalah 

begin mathsize 14px style D subscript f equals open curly brackets x vertical line x not equal to negative 3 space atau space x not equal to 3 comma space x element of straight R close curly brackets end style 

begin mathsize 14px style g left parenthesis x right parenthesis equals 2 x end style terdefinisi untuk setiap nilai begin mathsize 14px style x end style anggota bilangan real, Dengan demikian, daerah asal begin mathsize 14px style g open parentheses x close parentheses end style adalah 

begin mathsize 14px style D subscript g equals open curly brackets x vertical line x element of straight R close curly brackets end style 

sehingga 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f plus g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses x close parentheses plus g open parentheses x close parentheses end cell row blank equals cell fraction numerator 2 over denominator x squared minus 9 end fraction plus 2 x end cell row blank equals cell fraction numerator 2 over denominator x squared minus 9 end fraction plus fraction numerator 2 x open parentheses x squared minus 9 close parentheses over denominator x squared minus 9 end fraction end cell row blank equals cell fraction numerator 2 x cubed minus 18 x plus 2 over denominator x squared minus 9 end fraction end cell end table end style 

maka daerah asal fungsi begin mathsize 14px style left parenthesis f plus g right parenthesis left parenthesis x right parenthesis end style adalah 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript f intersection D subscript g end cell equals cell open curly brackets x vertical line x not equal to plus-or-minus 3 comma space x element of straight R close curly brackets intersection open curly brackets x vertical line space x element of straight R close curly brackets end cell row blank equals cell open curly brackets x vertical line x not equal to plus-or-minus 3 comma space x element of straight R close curly brackets end cell row blank equals cell open curly brackets x vertical line x not equal to negative 3 space atau space x not equal to negative 3 comma space x element of straight R close curly brackets end cell end table end style 

Jadi, jawaban yang tepat adalah D.

Roboguru

Diketahui fungsi , , dan . Tentukan nilai setiap fungsi berikut ini serta daerah asalnya (domain). a.

Pembahasan Soal:

Penjumlahan kedua fungsi f(x) dan g(x) adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f plus g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator 2 x over denominator x plus 1 end fraction plus fraction numerator x minus 1 over denominator 4 x end fraction end cell row blank equals cell fraction numerator open parentheses 2 x close parentheses open parentheses 4 x close parentheses plus open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses over denominator open parentheses x plus 1 close parentheses open parentheses 4 x close parentheses end fraction end cell row blank equals cell fraction numerator 8 x squared plus x squared minus 1 over denominator open parentheses x plus 1 close parentheses open parentheses 4 x close parentheses end fraction end cell row blank equals cell fraction numerator 9 x squared minus 1 over denominator open parentheses x plus 1 close parentheses open parentheses 4 x close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses 3 x plus 1 close parentheses open parentheses 3 x minus 1 close parentheses over denominator open parentheses x plus 1 close parentheses open parentheses 4 x close parentheses end fraction end cell end table 

  • f open parentheses x close parentheses equals fraction numerator 2 x over denominator x plus 1 end fraction terdefinisi apabila nilai x not equal to negative 1 maka D subscript f equals open curly brackets x vertical line space x not equal to negative 1 comma space x element of straight real numbers close curly brackets 
  • g open parentheses x close parentheses equals fraction numerator x minus 1 over denominator 4 x end fraction terdefinisi apabila nilai x not equal to 0 maka D subscript g equals open curly brackets x vertical line space x not equal to 0 comma space x element of straight real numbers close curly brackets 

Daerah asal fungsi open parentheses f plus g close parentheses open parentheses x close parentheses:

 table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript f plus g end subscript end cell equals cell D subscript f intersection D subscript g end cell row blank equals cell open curly brackets x vertical line space x not equal to 1 space comma space x element of straight real numbers close curly brackets intersection open curly brackets x space vertical line x not equal to 0 space comma space x element of straight real numbers close curly brackets end cell row blank equals cell open curly brackets x vertical line space x not equal to 1 space dan space x not equal to 0 comma space x element of straight real numbers close curly brackets end cell end table 

Roboguru

Diketahui  dan . Jika , daerah asal  adalah ....

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis end cell equals cell left parenthesis f plus g right parenthesis left parenthesis x right parenthesis end cell row blank equals cell fraction numerator 2 x plus 1 over denominator x minus 3 end fraction plus fraction numerator x plus 2 over denominator 7 end fraction end cell row blank equals cell fraction numerator 7 left parenthesis 2 x plus 1 right parenthesis plus left parenthesis x plus 2 right parenthesis left parenthesis x minus 3 right parenthesis over denominator left parenthesis x minus 3 right parenthesis 7 end fraction end cell row blank equals cell fraction numerator 14 x plus 7 plus x squared minus 3 x plus 2 x minus 6 over denominator 7 x minus 21 end fraction end cell row blank equals cell fraction numerator x squared plus 13 x plus 1 over denominator 7 x minus 21 end fraction end cell row blank blank blank row cell 7 x minus 21 end cell not equal to 0 row cell 7 x end cell not equal to 21 row x not equal to 3 end table end style

Jadi, jawaban yang tepat adalah E.

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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