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Diberikan dua buah fungsi f(x)=(x2−4)​ dan g(x)=2x−3. Tentukan rumus fungsi f+g, f⋅g, dan gf​, serta selanjutnya tentukan domain dari fungsi yang baru terbentuk!

Pertanyaan

Diberikan dua buah fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals square root of left parenthesis x squared minus 4 right parenthesis end root end style dan begin mathsize 14px style g left parenthesis x right parenthesis equals 2 x minus 3 end style.

Tentukan rumus fungsi undefinedundefined, dan undefined, serta selanjutnya tentukan domain dari fungsi yang baru terbentuk!

S. Difhayanti

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. Dr. Hamka

Jawaban terverifikasi

Pembahasan

Ingat!

Fungsi adalah pemetaan setiap anggota sebuah himpunan (yang dinamakan sebagai domain atau daerah asal) kepada anggota himpunan yang lain (yang dinamakan sebagai kodomain atau daerah kawan).

Operasi pada dua fungsi seperti berikut.

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 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 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 end table

Sehingga, diperoleh perhitungan berikut.

f left parenthesis x right parenthesis equals square root of open parentheses x squared minus 4 close parentheses end root straight D subscript f equals left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis  g left parenthesis x right parenthesis equals 2 x minus 3 straight D subscript g equals left parenthesis negative infinity comma infinity right parenthesis

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 square root of open parentheses x squared minus 4 close parentheses end root plus 2 x minus 3 end cell row cell straight D subscript f plus g end subscript end cell equals cell straight D subscript f intersection straight D subscript g end cell row cell straight D subscript f plus g end subscript end cell equals cell left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis intersection left parenthesis negative infinity comma infinity right parenthesis end cell row cell straight D subscript f plus g end subscript end cell equals cell left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis end cell end table

Fungsi table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f plus g close parentheses open parentheses x close parentheses end cell end table tidak mempunyai batas domain, sehingga domain fungsi tersebut negative infinity less than x less than infinity.

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f times g right parenthesis 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 cell left parenthesis f times g right parenthesis left parenthesis x right parenthesis end cell equals cell open parentheses square root of open parentheses x squared minus 4 close parentheses end root close parentheses times open parentheses 2 x minus 3 close parentheses end cell row cell straight D subscript f times g end subscript end cell equals cell straight D subscript f intersection straight D subscript g end cell row cell straight D subscript f times g end subscript end cell equals cell left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis intersection left parenthesis negative infinity comma infinity right parenthesis end cell row cell straight D subscript f times g end subscript end cell equals cell left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis end cell end table

Fungsi table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f times g close parentheses open parentheses x close parentheses end cell end table  tidak mempunyai batas domain, sehingga domain fungsi tersebut negative infinity less than x less than infinity.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f over g close parentheses left parenthesis x right parenthesis end cell equals cell fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction end cell row cell open parentheses f over g close parentheses left parenthesis x right parenthesis end cell equals cell fraction numerator square root of open parentheses x squared minus 4 close parentheses end root over denominator open parentheses 2 x minus 3 close parentheses end fraction end cell row cell straight D subscript open parentheses f over g close parentheses end subscript end cell equals cell straight D subscript f intersection straight D subscript g minus open curly brackets x equals 3 over 2 close curly brackets end cell row cell straight D subscript open parentheses f over g close parentheses end subscript end cell equals cell left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis intersection left parenthesis negative infinity comma infinity right parenthesis minus open curly brackets x equals 3 over 2 close curly brackets end cell row cell straight D subscript open parentheses f over g close parentheses end subscript end cell equals cell left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis minus open curly brackets x equals 3 over 2 close curly brackets end cell row cell straight D subscript open parentheses f over g close parentheses end subscript end cell equals cell left parenthesis negative infinity comma negative 3 right parenthesis union left parenthesis 3 comma infinity right parenthesis end cell end table

Fungsi table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f over g close parentheses open parentheses x close parentheses end cell end table memiliki batas-batas yaitu 2 x minus 3 greater than 0 rightwards double arrow 2 x greater than 3 rightwards double arrow x greater than 3 over 2 dan 2 x minus 3 less than 0 rightwards double arrow 2 x less than 3 rightwards double arrow x less than 3 over 2 , sehingga domain fungsi tersebut negative infinity less than 3 over 2 space dan space 3 over 2 less than x less than infinity.

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Grace Amelia Nathania ButarButar

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