Diketahui fungsi f(x)=x2−4​ dan g(x)=x−2​. Tentukanlah fungsi-fungsi berikut dan tentukan pula daerah asalnya. b)   (f−g)(x)

Pertanyaan

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

b)     

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

fungsi open parentheses f minus g close parentheses open parentheses x close parentheses equals square root of x minus 2 end root open parentheses square root of x plus 2 end root minus 1 close parentheses dan daerah asalnya adalah D subscript open parentheses f minus 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

Pembahasan

b) Diketahui fungsi  dan , maka diperoleh fungsi open parentheses f minus 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 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 square root of x squared minus 4 end root minus 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 minus 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 plus 2 end root minus square root of x minus 2 end root end cell row blank equals cell square root of x minus 2 end root open parentheses square root of x plus 2 end root minus 1 close parentheses 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 minus 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 minus 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 minus g close parentheses open parentheses x close parentheses equals square root of x minus 2 end root open parentheses square root of x plus 2 end root minus 1 close parentheses dan daerah asalnya adalah D subscript open parentheses f minus 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.

100

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