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​ dan g(x)=2x+6​1​

Pertanyaan

Tentukan rumus fungsi 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, jika diketahui begin mathsize 14px style f open parentheses x close parentheses end style dan begin mathsize 14px style g open parentheses x close parentheses end style berikut:

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  

M. Mariyam

Master Teacher

Mahasiswa/Alumni Institut Pertanian Bogor

Jawaban terverifikasi

Pembahasan

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 

 

 

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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−x−8​ dan g(x)=x+8​1​

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