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Jika diketahui dan , daerah asal (domain) fungsi adalah ...

Jika diketahui begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 2 over denominator x squared minus 9 end fraction end style dan begin mathsize 14px style g open parentheses x close parentheses equals 2 x end style, daerah asal (domain) fungsi begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses end style adalah ...

  1. begin mathsize 14px style open curly brackets x vertical line x not equal to negative 3 comma space x element of straight real numbers close curly brackets end style

  2. begin mathsize 14px style open curly brackets x vertical line x not equal to 3 comma space x element of straight real numbers close curly brackets end style

  3. begin mathsize 14px style left curly bracket x vertical line x not equal to 9 comma space x element of straight real numbers right curly bracket end style

  4. begin mathsize 14px style open curly brackets x vertical line x not equal to negative 3 space atau space x not equal to 3 comma space x element of straight real numbers close curly brackets end style

  5. begin mathsize 14px style open curly brackets x vertical line x not equal to negative 9 space atau space x not equal to 9 comma space x element of straight real numbers close curly brackets end style

Jawaban:

Perhatikan perhitungan berikut ini.

Ingat, penjumlahan dua fungsi:

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

Maka:

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 open parentheses fraction numerator 2 over denominator x squared minus 9 end fraction close parentheses plus open parentheses 2 x close parentheses end cell row blank equals cell fraction numerator 2 over denominator x squared minus 9 end fraction plus 2 x end cell row blank equals cell fraction numerator 2 plus 2 x open parentheses x squared minus 9 close parentheses over denominator x squared minus 9 end fraction end cell row blank equals cell fraction numerator 2 plus 2 x cubed minus 18 x over denominator x squared minus 9 end fraction end cell end table end style

Perhatikan bahwa pada fungsi rasional begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator p open parentheses x close parentheses over denominator q open parentheses x close parentheses end fraction end style, penyebut begin mathsize 14px style q open parentheses x close parentheses not equal to 0 end style, maka nilai undefined membuat penyebut sama dengan begin mathsize 14px style 0 end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus 9 end cell equals 0 row cell x squared end cell equals 9 row x equals cell plus-or-minus square root of 9 end cell row x equals cell plus-or-minus 3 end cell end table end style

Didapatkan nilai begin mathsize 14px style x equals 3 end style atau begin mathsize 14px style x equals negative 3 end style membuat penyebut sama dengan begin mathsize 14px style 0 end style, sehingga domainnya:

begin mathsize 14px style open curly brackets x vertical line x equals 3 space atau space x equals negative 3 comma space x element of straight real numbers close curly brackets end style

Jadi, domain fungsi begin mathsize 14px style open parentheses f plus g close parentheses open parentheses x close parentheses end style adalah begin mathsize 14px style open curly brackets x vertical line x equals 3 space atau space x equals negative 3 comma space x element of straight real numbers close curly brackets end style.

Dengan demikian, jawaban yang tepat adalah D.

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