Roboguru

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

Pertanyaan

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.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. Hajrianti

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 06 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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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.

1

Roboguru

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

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 cross times g close parentheses :

table attributes columnalign right center left columnspacing 2px 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 open parentheses x plus 3 close parentheses open parentheses x minus 9 close parentheses end cell row blank equals cell x squared plus 3 x minus 9 x minus 27 end cell row blank equals cell x squared minus 6 x minus 27 end cell end table   

Jadi open parentheses f cross times g close parentheses open parentheses x close parentheses equals x squared minus 6 x minus 27 dengan daerah asal D f equals open curly brackets x space vertical line space x element of R close curly brackets 

0

Roboguru

Tentukan rumus fungsi  dan daerah asalnya, jika diketahui  dan  berikut:

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 fungsi  dan . Tentukan hasil operasi aljabar fungsi berikut dan daerah asal hasil operasinya. b.

Pembahasan Soal:

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 perkalian tiga fungsi dapat dinotasikan:

begin mathsize 14px style open parentheses f times g times h close parentheses left parenthesis x right parenthesis equals f left parenthesis x right parenthesis times g left parenthesis x right parenthesis times h 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 perkalian dari tiga begin mathsize 14px style open parentheses f times g times h close parentheses left parenthesis x right parenthesis end style fungsi  adalah sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f times g times h close parentheses left parenthesis x right parenthesis end cell equals cell open parentheses fraction numerator 3 over denominator x minus 2 end fraction close parentheses times open parentheses 10 x squared minus 4 close parentheses times open parentheses square root of x squared minus 25 end root close parentheses end cell row blank equals cell fraction numerator 3 open parentheses 10 x squared minus 4 close parentheses open parentheses square root of x squared minus 25 end root close parentheses over denominator x minus 2 end fraction end cell end table end style 

Daerah asal dari perkalian tiga fungsi begin mathsize 14px style open parentheses f times g times h close parentheses left parenthesis x right parenthesis end style adalah:

Syarat dalam bentuk akar adalah nilai di dalam akar tidak boleh bernilai negatif, maka:

begin mathsize 14px style space space space space space space space space space x squared minus 25 greater or equal than 0 left parenthesis x minus 5 right parenthesis left parenthesis x plus 5 right parenthesis greater or equal than 0 x greater or equal than 5 space atau space x less or equal than negative 5 end style

Syarat dalam bentuk pecahan adalah nilai penyebut tidak boleh bernilai begin mathsize 14px style 0 end style, maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 2 end cell not equal to 0 row x not equal to 2 end table end style 

Sehingga, daerah asal dari perkalian tiga fungsi begin mathsize 14px style open parentheses f times g times h 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 less or equal than negative 5 space atau space x greater or equal than 5 close curly brackets end style

Jadi, hasil operasi perkalian tiga fungsi begin mathsize 14px style open parentheses f times g times h close parentheses left parenthesis x right parenthesis end style adalah begin mathsize 14px style open parentheses f times g times h close parentheses left parenthesis x right parenthesis equals fraction numerator 3 open parentheses 10 x squared minus 4 close parentheses open parentheses square root of x squared minus 25 end root close parentheses 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 less or equal than negative 5 space atau space x greater or equal than 5 close curly brackets end style.

0

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.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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