Jika f(x)=x2−5x+4 dan (g∘f)(x)=x+3, maka nilai g(−2) adalah ...

Pertanyaan

Jika $f(x)=x^2-5x+4$ dan $\left(g\circ f\right)\left(x\right)=x+3$ , maka nilai $g\left(-2\right)$ adalah ...

D. Nuryani

Master Teacher

Mahasiswa/Alumni Universitas Padjadjaran

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Diketahui f left parenthesis x right parenthesis equals x squared minus 5 x plus 4 dan open parentheses g ring operator f close parentheses open parentheses x close parentheses equals x plus 3 , maka fungsi g left parenthesis x right parenthesis adalah:

Misalkan x squared minus 5 x plus 4 equals p maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus 5 x plus 4 end cell equals p row cell x squared minus 5 x plus 4 minus p end cell equals 0 end table

Sehingga nilai x menggunakan rumus kuadratik adalah:

$table attributes columnalign right center left columnspacing 0px end attributes row cell x subscript 1 comma 2 end subscript end cell equals cell fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction end cell row cell x subscript 1 comma 2 end subscript end cell equals cell fraction numerator 5 plus-or-minus square root of 25 minus 4 open parentheses 4 minus p close parentheses end root over denominator 2 end fraction end cell row cell x subscript 1 comma 2 end subscript end cell equals cell fraction numerator 5 plus-or-minus square root of 25 minus 16 plus 4 p end root over denominator 2 end fraction end cell row cell x subscript 1 comma 2 end subscript end cell equals cell fraction numerator 5 plus-or-minus square root of 9 plus 4 p end root over denominator 2 end fraction end cell end table$

Fungsi g left parenthesis x right parenthesis menjadi:

$table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x squared minus 5 x plus 4 right parenthesis end cell equals cell x plus 3 space space space space space space space space space space space space space space space space space atau space space g left parenthesis x squared minus 5 x plus 4 right parenthesis equals x plus 3 end cell row cell g left parenthesis p right parenthesis end cell equals cell fraction numerator 5 plus square root of 9 plus 4 p end root over denominator 2 end fraction plus 3 space space space space space space space space space space space space space space space space space space space space space space space g left parenthesis p right parenthesis equals fraction numerator 5 minus square root of 9 plus 4 p end root over denominator 2 end fraction plus 3 end cell row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 5 plus square root of 9 plus 4 x end root over denominator 2 end fraction plus 3 space space space space space space space space space space space space space space space space space space space space space space space g left parenthesis x right parenthesis equals fraction numerator 5 minus square root of 9 plus 4 x end root over denominator 2 end fraction plus 3 end cell row blank blank blank end table$

Ditanyakan nilai dari g open parentheses negative 2 close parentheses , yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses negative 2 close parentheses end cell equals cell fraction numerator 5 plus square root of 9 plus 4 open parentheses negative 2 close parentheses end root over denominator 2 end fraction plus 3 space atau space g left parenthesis negative 2 right parenthesis equals fraction numerator 5 minus square root of 9 plus 4 open parentheses negative 2 close parentheses end root over denominator 2 end fraction plus 3 end cell row cell g open parentheses negative 2 close parentheses end cell equals cell fraction numerator 5 plus square root of 9 minus 8 end root space over denominator 2 end fraction plus 3 space space space space space space space space space space space space space space space space space g open parentheses negative 2 close parentheses equals fraction numerator 5 minus square root of 9 minus 8 end root space over denominator 2 end fraction plus 3 end cell row cell g left parenthesis negative 2 right parenthesis end cell equals cell fraction numerator 5 plus square root of 1 over denominator 2 end fraction plus 3 space space space space space space space space space space space space space space space space space space space space space space space g left parenthesis negative 2 right parenthesis equals fraction numerator 5 minus square root of 1 over denominator 2 end fraction plus 3 end cell row cell g left parenthesis negative 2 right parenthesis end cell equals cell 6 over 2 plus 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space g left parenthesis negative 2 right parenthesis equals 4 over 2 plus 3 end cell row cell g left parenthesis negative 2 right parenthesis end cell equals cell 6 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space g left parenthesis negative 2 right parenthesis equals 5 end cell end table$

Dengan demikian, nilai dari g open parentheses negative 2 close parentheses adalah 6 space atau space 5 .

Oleh karena itu, jawaban yang benar adalah D.