Roboguru

Diketahui fungsi  dan . Tentukanlah fungsi-fungsi berikut dan tentukan pula daerah asalnya. c)

Pertanyaan

Diketahui fungsi  dan . Tentukanlah fungsi-fungsi berikut dan tentukan pula daerah asalnya.

c)     

Pembahasan Soal:

c) Diketahui fungsi  dan , maka diperoleh fungsi open parentheses f cross times g close parentheses open parentheses x close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f cross 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 4 end root times square root of x minus 2 end root end cell row blank equals cell square root of open parentheses x minus 2 close parentheses open parentheses x plus 2 close parentheses end root times square root of x minus 2 end root end cell row blank equals cell open parentheses square root of x minus 2 end root square root of x plus 2 end root close parentheses times square root of x minus 2 end root end cell row blank equals cell square root of x minus 2 end root square root of x minus 2 end root square root of x plus 2 end root end cell row blank equals cell open parentheses x minus 2 close parentheses square root of x plus 2 end root end cell end table
 

Kemudian, fungsi f open parentheses x close parentheses equals square root of x squared minus 4 end root terdefinisi saat x squared minus 4 greater or equal than 0. Nilai x yang memenuhi x squared minus 4 greater or equal than 0 sebagai berikut.

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 minus 2 close parentheses open parentheses x plus 2 close parentheses end cell greater or equal than 0 end table

Berdasarkan bentuk diatas, maka x squared minus 4 greater or equal than 0 saat x minus 2 greater or equal than 0 dan x plus 2 greater or equal than 0 atau x minus 2 less or equal than 0 dan x plus 2 less or equal than 0. Saat x minus 2 greater or equal than 0 dan x plus 2 greater or equal than 0, nilai x yang memenuhi sebagai berikut.

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

table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 2 end cell greater or equal than 0 row x greater or equal than cell negative 2 end cell end table

x greater or equal than 2 dan x greater or equal than negative 2, maka nilai x yang memenuhi keduanya adalah x greater or equal than 2.

Lalu, saat x minus 2 less or equal than 0 dan x plus 2 less or equal than 0, nilai x yang memenuhi sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 2 end cell less or equal than 0 row x less or equal than 2 end table
 

table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 2 end cell less or equal than 0 row x less or equal than cell negative 2 end cell end table

x less or equal than 2 dan x less or equal than negative 2, maka nilai x yang memenuhi keduanya adalah x less or equal than negative 2.

 x squared minus 4 greater or equal than 0 saat x less or equal than negative 2 atau x greater or equal than 2, maka daerah asal fungsi f open parentheses x close parentheses tersebut adalah D subscript f equals open curly brackets x vertical line space x less or equal than negative 2 space atau space x greater or equal than 2 comma space x element of R close curly brackets.

Selanjutnya, fungsi g open parentheses x close parentheses equals square root of x minus 2 end root terdefinisi saat x minus 2 greater or equal than 0x minus 2 greater or equal than 0 atau x greater or equal than 2, maka daerah asal fungsi g open parentheses x close parentheses tersebut adalah D subscript g equals open curly brackets x vertical line space x greater or equal than 2 comma space x element of R close curly brackets.

Berdasarkan daerah asal fungsi f open parentheses x close parentheses dan g open parentheses x close parentheses, maka daerah asal fungsi open parentheses f cross times g close parentheses open parentheses x close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript open parentheses f cross times g close parentheses 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 less or equal than negative 2 space atau space x greater or equal than 2 comma space x element of R close curly brackets space intersection space open curly brackets x vertical line space x greater or equal than 2 comma space x element of R close curly brackets end cell row blank equals cell open curly brackets x vertical line space x greater or equal than 2 comma space x element of R close curly brackets end cell end table


Dengan demikian, fungsi open parentheses f cross times g close parentheses open parentheses x close parentheses equals open parentheses x minus 2 close parentheses square root of x plus 2 end root dan daerah asalnya adalah D subscript open parentheses f cross times g close parentheses end subscript equals open curly brackets x vertical line space x greater or equal than 2 comma space x element of R close curly brackets.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. Hajrianti

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 05 Juni 2021

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Pertanyaan yang serupa

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

Pembahasan Soal:

Diketahui fungsi f open parentheses x close parentheses equals fraction numerator x plus 4 over denominator x minus 5 end fraction dan g open parentheses x close parentheses equals square root of x squared minus 9 x end root. Jika h open parentheses x close parentheses equals open parentheses f cross times g close parentheses open parentheses x close parentheses maka:

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 cross times g close parentheses open parentheses x close parentheses end cell row blank equals cell f open parentheses x close parentheses cross times g open parentheses x close parentheses end cell row blank equals cell fraction numerator x plus 4 over denominator x minus 5 end fraction open parentheses square root of x squared minus 9 x end root close parentheses end cell row blank equals cell fraction numerator open parentheses x plus 4 close parentheses square root of x squared minus 9 x end root over denominator x minus 5 end fraction end cell end table 

Supaya h open parentheses x close parentheses terdifinis maka penyebut tidak boleh sama dengan nol, sehingga pembuat nolnya adalah:

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 5 end cell equals 0 row x equals 5 end table

sehingga x not equal to 5, kemudian supaya square root of x squared minus 9 x end root terdefinisi maka:

x squared minus 9 x greater or equal than 0 x open parentheses x minus 9 close parentheses greater or equal than 0

diperoleh x greater or equal than 0 atau x greater or equal than 9, oleh karena jika x equals 1 diperoleh x squared minus 9 x less than 0 maka tidak memenuhi. 

Dengan demikian daerah asal fungsi h open parentheses x close parentheses adalah open curly brackets x vertical line x less or equal than 0 space atau space x greater or equal than 9 comma space x element of straight R close curly brackets.

Oleh karena itu, jawaban yang benar adalah D.

Roboguru

Diketahui fungsi  dan . Tentukanlah fungsi-fungsi berikut dan tentukan pula daerah asalnya. c.

Pembahasan Soal:

Hasil dari fungsi begin mathsize 14px style open parentheses f cross times g close parentheses open parentheses x close parentheses end style didapatkan:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f cross times g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses x close parentheses cross times g open parentheses x close parentheses end cell row blank equals cell square root of x squared minus 4 end root cross times square root of x minus 2 end root end cell row blank equals cell square root of open parentheses x squared minus 4 close parentheses open parentheses x minus 2 close parentheses end root end cell row blank equals cell square root of x cubed minus 2 x squared minus 4 x plus 8 end root end cell end table end style 

Daerah asalnya dari fungsi begin mathsize 14px style open parentheses f cross times g close parentheses open parentheses x close parentheses end style didapatkan:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x cubed minus 2 x squared minus 4 x plus 8 end cell greater or equal than 0 row cell open parentheses x minus 2 close parentheses open parentheses x minus 2 close parentheses open parentheses x plus 2 close parentheses end cell greater or equal than 0 end table end style 

begin mathsize 14px style x equals 2 comma space x equals negative 2 end style 

Sehingga didapatkan:



Sehingga , daerah asalnya dari fungsi begin mathsize 14px style open parentheses f cross times g close parentheses open parentheses x close parentheses end style adalah begin mathsize 14px style x greater or equal than negative 2 end style.

Roboguru

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

Pembahasan Soal:

Jika terdapat fungsi begin mathsize 14px style f left parenthesis x right parenthesis space dan space g left parenthesis x right parenthesis end style dimana begin mathsize 14px style open parentheses f times g close parentheses left parenthesis x right parenthesis end style maka didefinisikan sebagai begin mathsize 14px style open parentheses f times g close parentheses left parenthesis x right parenthesis equals f left parenthesis x right parenthesis times g left parenthesis x right parenthesis end style dengan daerah asal begin mathsize 14px style D subscript f times g end subscript equals D subscript f intersection D subscript g end style

Diketahui:

 begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator x plus 4 over denominator x minus 5 end fraction end style

begin mathsize 14px style g left parenthesis x right parenthesis equals square root of x squared minus 9 x end root end style

Hasil kali dua fungsi begin mathsize 14px style open parentheses f times g 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 f times g close parentheses left parenthesis x right parenthesis end cell equals cell f left parenthesis x right parenthesis times g left parenthesis x right parenthesis end cell row blank equals cell fraction numerator x plus 4 over denominator x minus 5 end fraction times square root of x squared minus 9 x end root end cell row blank equals cell fraction numerator open parentheses x plus 4 close parentheses open parentheses square root of x squared minus 9 x end root close parentheses over denominator x minus 5 end fraction end cell end table end style

Sehingga, domain hasil kali dua fungsi tersebut adalah begin mathsize 14px style open curly brackets x space left enclose space x less or equal than 0 space atau space x greater or equal than 9 comma space x element of R end enclose close curly brackets end style

Jadi, jawaban yang tepat adalah D.

Roboguru

Diketahui fungsi ,  dan . Tentukan rumus fungsi berikut apabila terdefinisi dan tentukan daerah asal dan daerah hasilnya. c)

Pembahasan Soal:

Diketahui fungsi  dan , maka fungsi open parentheses f cross times g close parentheses open parentheses x close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f cross 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 open parentheses fraction numerator x minus 3 over denominator x end fraction close parentheses times square root of x squared minus 9 end root end cell row blank equals cell fraction numerator open parentheses x minus 3 close parentheses square root of x squared minus 9 end root over denominator x end fraction end cell end table

Fungsi f open parentheses x close parentheses equals fraction numerator x minus 3 over denominator x end fraction terdefinisi saat x not equal to 0, maka D subscript f equals open curly brackets x vertical line space x not equal to 0 comma space x element of straight real numbers close curly brackets. Kemudian, fungsi g open parentheses x close parentheses equals square root of x squared minus 9 end root terdefinisi saat x squared minus 9 greater or equal than 0. Nilai x yang memenuhi saat x squared minus 9 greater or equal than 0 sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus 9 end cell greater or equal than 0 row cell open parentheses x minus 3 close parentheses open parentheses x plus 3 close parentheses end cell greater or equal than 0 end table

Berdasarkan uraian di atas, x squared minus 9 greater or equal than 0 saat x minus 3 greater or equal than 0 dan x plus 3 greater or equal than 0 atau x minus 3 less or equal than 0 atau x plus 3 less or equal than 0. Saat x minus 3 greater or equal than 0 dan x plus 3 greater or equal than 0, nilai x yang memenuhi sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 3 end cell greater or equal than 0 row x greater or equal than 3 end table


table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 3 end cell greater or equal than 0 row x greater or equal than cell negative 3 end cell end table

x greater or equal than 3 dan x greater or equal than negative 3, maka nilai x yang memenuhi keduanya adalah x greater or equal than 3. Lalu, saat x minus 3 less or equal than 0 atau x plus 3 less or equal than 0, nilai x yang memenuhi sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 3 end cell less or equal than 0 row x less or equal than 3 end table


table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 3 end cell less or equal than 0 row x less or equal than cell negative 3 end cell end table

x less or equal than 3 dan x less or equal than negative 3, maka nilai x yang memenuhi keduanya adalah x less or equal than negative 3.

Berdasarkan uraian di atas, x squared minus 9 greater or equal than 0 saat x greater or equal than 3 atau x less or equal than negative 3, maka D subscript g equals open curly brackets x vertical line space x less or equal than negative 3 space atau space x greater or equal than 3 comma space x element of straight real numbers close curly brackets.

Berdasarkan daerah asal fungsi f dan fungsi g di atas, maka diperoleh D subscript open parentheses f cross times g close parentheses end subscript atau daerah asal fungsi open parentheses f cross times g close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript open parentheses f cross times g close parentheses 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 0 comma space x element of straight real numbers close curly brackets intersection open curly brackets x vertical line x less or equal than negative 3 space atau space x greater or equal than 3 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 less or equal than negative 3 space atau space x greater or equal than 3 comma space x element of straight real numbers close curly brackets end cell end table


Selanjutnya, daerah hasil fungsi open parentheses f cross times g close parentheses open parentheses x close parentheses di atas adalah open curly brackets y vertical line space y greater or equal than 0 comma space y element of straight real numbers close curly brackets

Dengan demikian, fungsi open parentheses f cross times g close parentheses adalah open parentheses f cross times g close parentheses open parentheses x close parentheses equals fraction numerator x minus 3 over denominator x end fraction plus square root of x squared minus 9 end root, daerah asalnya adalah D subscript open parentheses f cross times g close parentheses end subscript equals open curly brackets x vertical line space x less or equal than negative 3 space atau space x greater or equal than 3 comma space x element of straight real numbers close curly brackets, dan daerah hasilnya adalahR subscript open parentheses f cross times g close parentheses end subscript equals open curly brackets y vertical line space y greater or equal than 0 comma space y element of straight real numbers close curly brackets.

Roboguru

Diketahui  dan . Tentukan rumus fungsi dan daerah asal (domain) dari setiap fungsi berikut.

Pembahasan Soal:

Diketahui begin mathsize 14px style f open parentheses x close parentheses equals x minus 4 end style dan begin mathsize 14px style g open parentheses x close parentheses equals 2 x squared minus 5 x minus 12 end style sehingga diperoleh begin mathsize 14px style open parentheses f times g close parentheses open parentheses x close parentheses end style sebagai berikut.

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 open parentheses x minus 4 close parentheses open parentheses 2 x squared minus 5 x minus 12 close parentheses end cell row blank equals cell 2 x cubed minus 5 x squared minus 12 x minus 8 x squared plus 20 x plus 48 end cell row blank equals cell 2 x cubed minus 5 x squared minus 8 x squared minus 12 x plus 20 x plus 48 end cell row blank equals cell 2 x cubed minus 13 x squared plus 8 x plus 48 end cell row blank blank blank end table end style

begin mathsize 14px style f open parentheses x close parentheses end style terdefinisi untuk setiap begin mathsize 14px style x element of R end style sehingga daerah asalnya  begin mathsize 14px style D subscript f equals left curly bracket x vertical line x element of R right curly bracket end style dan begin mathsize 14px style g open parentheses x close parentheses end style terdefinisi untuk setiap begin mathsize 14px style x element of R end stylesehingga daerah asalnya begin mathsize 14px style D subscript g equals left curly bracket x vertical line x element of R right curly bracket end style. Oleh karena itu daerah asal fungsi begin mathsize 14px style open parentheses f times g close parentheses open parentheses x close parentheses end style sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript open parentheses f times g close parentheses end subscript end cell equals cell D subscript f intersection D subscript g end cell row blank equals cell left curly bracket x vertical line x element of R right curly bracket space intersection space left curly bracket x vertical line x element of R right curly bracket end cell row blank equals cell left curly bracket x vertical line x element of R right curly bracket end cell end table end style 


Jadi, begin mathsize 14px style open parentheses f times g close parentheses open parentheses x close parentheses equals 2 x cubed minus 13 x squared plus 8 x plus 48 end style dan daerah asalnya yaitu begin mathsize 14px style D subscript open parentheses f times g close parentheses end subscript equals left curly bracket x vertical line x element of R right curly bracket end style.

Roboguru

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