Roboguru

Diketahui fungsi f(x)=x+3 dan g(x)=x−9. Tentukanlah fungsi-fungsi berikut dan tentukan pula daerah asalnya. (f−g)

Pertanyaan

Diketahui fungsi f left parenthesis x right parenthesis equals x plus 3 dan g left parenthesis x right parenthesis equals x minus 9. Tentukanlah fungsi-fungsi berikut dan tentukan pula daerah asalnya.

open parentheses f minus g close parentheses  
 

Pembahasan Soal:

Fungsi dapat mengalami operasi penjumlahan, perkalian, pembagian dan pengurangan seperti bilangan.

Bila ada dua buah fungsi p left parenthesis x right parenthesis spacedan q left parenthesis x right parenthesis berlaku :

table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses p plus q close parentheses open parentheses x close parentheses end cell equals cell p open parentheses x close parentheses plus q open parentheses x close parentheses end cell row cell open parentheses p minus q close parentheses open parentheses x close parentheses end cell equals cell p open parentheses x close parentheses minus q open parentheses x close parentheses end cell row cell open parentheses p divided by q close parentheses open parentheses x close parentheses end cell equals cell fraction numerator p open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction end cell row cell open parentheses p cross times q close parentheses open parentheses x close parentheses end cell equals cell p open parentheses x close parentheses cross times q open parentheses x close parentheses end cell end table 

Jika diketahui fungsi f left parenthesis x right parenthesis equals x plus 3 dan g left parenthesis x right parenthesis equals x minus 9, maka open parentheses f minus g close parentheses  :

table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses f minus g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses x close parentheses minus g open parentheses x close parentheses end cell row blank equals cell open parentheses x plus 3 close parentheses minus open parentheses x minus 9 close parentheses end cell row blank equals cell x plus 3 minus x plus 9 end cell row blank equals 12 end table  

Jadi open parentheses f minus g close parentheses open parentheses x close parentheses equals 12  

 

 

 

 

 

 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Terakhir diupdate 06 Oktober 2021

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Pertanyaan yang serupa

Diketahui fungsi f(x)=x−23​,g(x)=10x2−4, dan h(x)=x2−25​. Tentukan hasil operasi aljabar fungsi berikut dan daerah asal hasil operasinya. a. (f+g)(x)

Pembahasan Soal:

Ingat bahwa:

Fungsi merupakan suatu relasi yang memetakan setiap anggota dari suatu himpunan yang disebut sebagai daerah asal atau doamin ke tepat satu anggota himpunan lain yang disebut daerah kawan (kodomain).  

Operasi penjumlahan dua fungsi dapat dinotasikan:

begin mathsize 14px style open parentheses f plus g close parentheses left parenthesis x right parenthesis equals f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end style

Diketahui:

begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 3 over denominator x minus 2 end fraction g left parenthesis x right parenthesis equals 10 x squared minus 4 h left parenthesis x right parenthesis equals square root of x squared minus 25 end root end style 

Hasil operasi penjumlahan dari dua fungsi begin mathsize 14px style open parentheses f plus g close parentheses left parenthesis x right parenthesis end style adalah sebagai berikut:

begin mathsize 14px style 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 f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end cell row blank equals cell fraction numerator 3 over denominator x minus 2 end fraction plus 10 x squared minus 4 end cell row blank equals cell fraction numerator 3 plus left parenthesis 10 x squared minus 4 right parenthesis left parenthesis x minus 2 right parenthesis over denominator x minus 2 end fraction end cell row blank equals cell fraction numerator 3 plus 10 x cubed minus 20 x squared minus 4 x plus 8 over denominator x minus 2 end fraction end cell row blank equals cell fraction numerator 10 x cubed minus 20 x squared minus 4 x plus 11 over denominator x minus 2 end fraction end cell end table end style

Daerah asal dari penjumlahan dua fungsi begin mathsize 14px style open parentheses f plus g close parentheses left parenthesis x right parenthesis end style adalah begin mathsize 14px style open curly brackets x element of R space colon space x not equal to 2 close curly brackets end style.

Jadi, hasil operasi penjumlahan dua fungsi begin mathsize 14px style open parentheses f plus g close parentheses left parenthesis x right parenthesis end style adalah begin mathsize 14px style left parenthesis f plus g right parenthesis left parenthesis x right parenthesis equals fraction numerator 10 x cubed minus 20 x squared minus 4 x plus 11 over denominator x minus 2 end fraction end style dengan daerah asalnya adalah begin mathsize 14px style open curly brackets x element of R space colon space x not equal to 2 close curly brackets end style.

0

Roboguru

Fungsi f dan fungsi g ditentukan oleh rumus f(x)=x+11​ dan g(x)=x2−2. Tentukan fungsi-fungsi berikut, kemudian tentukan domain alaminya.  a. (f+g)(x)

Pembahasan Soal:

Gunakan konsep penjumlahan dua fungsi dan domain fungsi rasional.

begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses equals f open parentheses x close parentheses plus g open parentheses x close parentheses end style

Diketahui:

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

Akan ditentukan begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses end style dan domainnya.

Menentukan begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses end style, perhatikan perhitungan berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px 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 open parentheses fraction numerator 1 over denominator x plus 1 end fraction close parentheses plus open parentheses x squared minus 2 close parentheses end cell row blank equals cell fraction numerator 1 over denominator x plus 1 end fraction plus fraction numerator open parentheses x squared minus 2 close parentheses open parentheses x plus 1 close parentheses over denominator x plus 1 end fraction end cell row blank equals cell fraction numerator 1 plus open parentheses x cubed plus x squared minus 2 x minus 2 close parentheses over denominator x plus 1 end fraction end cell row cell open parentheses f plus g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator x cubed plus x squared minus 2 x minus 1 over denominator x plus 1 end fraction end cell end table end style  

Diperoleh begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses equals fraction numerator x cubed plus x squared minus 2 x minus 1 over denominator x plus 1 end fraction end style, yang merupakan bentuk rasional, syarat agar fungsi rasional terdefinisi adalah penyebutnya tidak boleh bernilai nol.

Perhatikan penyebutnya tidak boleh bernilai nol diperolehbegin mathsize 14px style x plus 1 not equal to 0 rightwards arrow x not equal to negative 1 end style, sehingga diperoleh domainnya adalah begin mathsize 14px style open curly brackets x vertical line x not equal to negative 1 comma space x element of straight real numbers close curly brackets end style.

Jadi, diperoleh begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses equals fraction numerator x cubed plus x squared minus 2 x minus 1 over denominator x plus 1 end fraction end style dan domain alaminya adalah begin mathsize 14px style open curly brackets x vertical line x not equal to negative 1 comma space x element of straight real numbers close curly brackets end style.

0

Roboguru

Tentukan rumus fungsi (f+g)(x),(f⋅g)(x),(gf​)(x) dan daerah asalnya, jika diketahui f(x) dan g(x) berikut: f(x)=x2−16​dang(x)=2x+6​1​

Pembahasan Soal:

Diketahui : begin mathsize 14px style f left parenthesis x right parenthesis equals square root of x squared minus 16 end root space dan space g left parenthesis x right parenthesis equals fraction numerator 1 over denominator square root of 2 x plus 6 end root end fraction end style

Ditanyakan: begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses comma space open parentheses f times g close parentheses open parentheses x close parentheses comma space open parentheses f over g close parentheses open parentheses x close parentheses end style dan daerah asalnya

Jawab:

a.

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 square root of x squared minus 16 end root plus fraction numerator 1 over denominator square root of 2 x plus 6 end root end fraction end cell end table end style  

daerah asal begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses end style :

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript f end cell equals cell open curly brackets right enclose x space x squared minus 16 greater or equal than 0 close curly brackets end cell row blank equals cell open curly brackets right enclose x space open parentheses x plus 4 close parentheses open parentheses x minus 4 close parentheses greater or equal than 0 comma space x element of straight real numbers close curly brackets end cell row blank equals cell open curly brackets right enclose x space x less or equal than negative 4 space atau space x greater or equal than 4 comma space x element of straight real numbers close curly brackets end cell row cell D subscript g end cell equals cell open curly brackets right enclose x space 2 x plus 6 greater than 0 close curly brackets end cell row blank equals cell open curly brackets right enclose x space x greater than negative 3 comma space x element of straight real numbers close curly brackets end cell row blank blank blank end table end style 

begin mathsize 14px style 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 right enclose x space x less or equal than negative 4 space atau space x greater or equal than 4 comma space x element of straight real numbers close curly brackets intersection open curly brackets right enclose x space x greater than negative 3 comma space x element of straight real numbers close curly brackets end cell row blank equals cell open curly brackets right enclose x space x greater-than or slanted equal to 4 comma space x element of straight real numbers close curly brackets end cell end table end style  

b. 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f times g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses x close parentheses times g open parentheses x close parentheses end cell row blank equals cell square root of x squared minus 16 end root times fraction numerator 1 over denominator square root of 2 x plus 6 end root end fraction end cell row blank equals cell fraction numerator square root of x squared minus 16 end root over denominator square root of 2 x plus 6 end root end fraction times fraction numerator square root of 2 x plus 6 end root over denominator square root of 2 x plus 6 end root end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 x plus 6 end fraction square root of open parentheses x squared minus 16 close parentheses open parentheses 2 x plus 6 close parentheses end root end cell row blank equals cell fraction numerator 1 over denominator 2 x plus 6 end fraction square root of 2 x cubed plus 6 x squared minus 32 x minus 96 end root end cell end table end style  

daerah asal begin mathsize 14px style open parentheses f. g close parentheses open parentheses x close parentheses end style :

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript f end cell equals cell open curly brackets right enclose x space x squared minus 16 greater or equal than 0 close curly brackets end cell row blank equals cell open curly brackets right enclose x space open parentheses x plus 4 close parentheses open parentheses x minus 4 close parentheses greater or equal than 0 comma space x element of straight real numbers close curly brackets end cell row blank equals cell open curly brackets right enclose x space x less or equal than negative 4 space atau space x greater or equal than 4 comma space x element of straight real numbers close curly brackets end cell row cell D subscript g end cell equals cell open curly brackets right enclose x space 2 x plus 6 greater than 0 close curly brackets end cell row blank equals cell open curly brackets right enclose x space x greater than negative 3 comma space x element of straight real numbers close curly brackets end cell row blank blank blank end table end style  

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript f times g end subscript end cell equals cell D subscript f intersection D subscript g end cell row blank equals cell left curly bracket right enclose x space x less or equal than negative 4 space atau space x greater or equal than 4 comma space x element of straight real numbers right curly bracket intersection left curly bracket right enclose x space x greater than negative 3 comma space x element of straight real numbers right curly bracket end cell row blank equals cell left curly bracket right enclose x space x greater-than or slanted equal to 4 comma space x element of straight real numbers right curly bracket end cell end table end style

c.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f over g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction end cell row blank equals cell square root of x squared minus 16 end root divided by fraction numerator 1 over denominator square root of 2 x plus 6 end root end fraction end cell row blank equals cell square root of x squared minus 16 end root times square root of 2 x plus 6 end root end cell row blank equals cell square root of 2 x cubed plus 6 x squared minus 32 x minus 96 end root end cell end table end style 

Daerah asal begin mathsize 14px style open parentheses f over g close parentheses open parentheses x close parentheses end style :

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript f end cell equals cell open curly brackets right enclose x space x squared minus 16 greater or equal than 0 close curly brackets end cell row blank equals cell open curly brackets right enclose x space open parentheses x plus 4 close parentheses open parentheses x minus 4 close parentheses greater or equal than 0 comma space x element of straight real numbers close curly brackets end cell row blank equals cell open curly brackets right enclose x space x less or equal than negative 4 space atau space x greater or equal than 4 comma space x element of straight real numbers close curly brackets end cell row cell D subscript g end cell equals cell open curly brackets right enclose x space 2 x plus 6 greater than 0 close curly brackets end cell row blank equals cell open curly brackets right enclose x space x greater than negative 3 comma space x element of straight real numbers close curly brackets end cell row blank blank blank end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript f over g end subscript end cell equals cell D subscript f intersection D subscript g space comma space g open parentheses x close parentheses not equal to 0 end cell row blank equals cell left curly bracket right enclose x space x less or equal than negative 4 space atau space x greater or equal than 4 comma space x element of straight real numbers right curly bracket intersection left curly bracket right enclose x space x greater than negative 3 comma space x element of straight real numbers right curly bracket comma space x not equal to negative 3 end cell row blank equals cell left curly bracket right enclose x space x greater-than or slanted equal to 4 comma space x element of straight real numbers right curly bracket end cell end table end style 

 

 

0

Roboguru

Diketahui f(x)=x−32x+1​ dan g(x)=7x+2​. Jika h(x)=(f+g)(x), daerah asal h(x) adalah ...

Pembahasan Soal:

Diketahui f open parentheses x close parentheses equals fraction numerator 2 x plus 1 over denominator x minus 3 end fraction dan g open parentheses x close parentheses equals fraction numerator x plus 2 over denominator 7 end fraction, berdasarkan konsep penjumlahan dua fungsi maka hasil dari fungsi h open parentheses x close parentheses equals open parentheses f plus g close parentheses open parentheses x close parentheses adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell h open parentheses x close parentheses end cell equals cell open parentheses f plus g close parentheses open parentheses x close parentheses end cell row blank equals cell f open parentheses x close parentheses plus g open parentheses x close parentheses 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 open parentheses 2 x plus 1 close parentheses over denominator 7 open parentheses x minus 3 close parentheses end fraction plus fraction numerator open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses over denominator 7 open parentheses x minus 3 close parentheses end fraction end cell row blank equals cell fraction numerator 14 x plus 7 over denominator 7 x minus 21 end fraction plus fraction numerator x squared minus 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 end table 

Fungsi h open parentheses x close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x squared plus 13 x plus 1 over denominator 7 x minus 21 end fraction end cell end table terdefinisi apabila penyebutnya tidak sama dengan nol, maka pembuat nolnya adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell 7 x minus 21 end cell equals 0 row cell 7 x end cell equals 21 row x equals 3 end table

Dengan demikian daerah asal h open parentheses x close parentheses adalah open curly brackets x vertical line x not equal to 3 comma space x element of straight R close curly brackets.

0

Roboguru

Diketahui fungsi-fungsi f dan g ditentukan dengan rumus f(x)=2x−10 dan g(x)=2x−1. Tentukan nilai fungsi-fungsi berikut, kemudian tentukan daerah asalnya. 1. (f+g)(x)

Pembahasan Soal:

Perlu diingat bahwa left parenthesis f plus g right parenthesis left parenthesis x right parenthesis equals f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis dengan daerah asal D subscript f plus g end subscript equals D subscript f intersection D subscript g, sehingga:

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 2 x minus 10 plus 2 x minus 1 end cell row blank equals cell 2 x plus 2 x minus 10 minus 1 end cell row blank equals cell 4 x minus 11 end cell end table

Berdasarkan uraian di atas, diperoleh open parentheses f plus g close parentheses open parentheses x close parentheses equals 4 x minus 11

Kemudian, domain (daerah asal) adalah himpunan semua bilangan real x sehingga suatu fungsi terdefinisi.  Fungsi f open parentheses x close parentheses equals 2 x minus 10 terdefinisi untuk semua x element of straight real numbers, maka D subscript f equals open curly brackets x vertical line space x element of straight real numbers close curly brackets. Lalu, fungsi g open parentheses x close parentheses equals 2 x minus 1 terdefinisi untuk semua x element of straight real numbers, maka D subscript g equals open curly brackets x vertical line space x element of straight real numbers close curly brackets. Berdasarkan daerah asal fungsi f open parentheses x close parentheses dan g open parentheses x close parentheses tersebut, diperoleh asal fungsi begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses space end style sebagai berikut.

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 element of straight real numbers close curly brackets intersection open curly brackets x vertical line space x element of straight real numbers close curly brackets end cell row blank equals cell open curly brackets x vertical line space x element of straight real numbers close curly brackets end cell end table


Dengan demikian, open parentheses f plus g close parentheses open parentheses x close parentheses equals 4 x minus 11 dan daerah asal begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses space end styleadalah D subscript open parentheses f plus g close parentheses end subscript equals open curly brackets x vertical line space x element of straight real numbers close curly brackets.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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