Misal N adalah himpunan bilangan asli, P adalah himpunan bilangan prima dan R adalah himpunan bilangan real. Jika f:P→N; g:N→R dan h:P→R dengan g(x)=x+1, f(x)=x−13x2−6x−27​ dan h=f∘g. Tentukanlah nilai  agar h(a)=9.

Pertanyaan

Misal N adalah himpunan bilangan asli, P adalah himpunan bilangan prima dan R adalah himpunan bilangan real. Jika f colon P rightwards arrow Ng colon N rightwards arrow R dan h colon P rightwards arrow R dengan g left parenthesis x right parenthesis equals x plus 1f left parenthesis x right parenthesis equals fraction numerator 3 x squared minus 6 x minus 27 over denominator x minus 1 end fraction dan h equals f ring operator g. Tentukanlah nilai a agar h left parenthesis a right parenthesis equals 9.

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Jawaban terverifikasi

Jawaban

nilai a adalah fraction numerator 9 plus 3 square root of 37 over denominator 6 end fraction atau fraction numerator 9 minus 3 square root of 37 over denominator 6 end fraction.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah fraction numerator 9 plus 3 square root of 37 over denominator 6 end fraction atau fraction numerator 9 minus 3 square root of 37 over denominator 6 end fraction.

Ingat!

  • Misalkan f colon y rightwards arrow z dan g colon x rightwards arrow y. Komposisi fungsi g dan f yang memetakan x ke y kemudian ke z adalah:

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell end table

  • Misalkan terdapat persamaan kuadrat a x squared plus b x plus c equals 0, maka 

x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction

Diketahui  g left parenthesis x right parenthesis equals x plus 1 dan f left parenthesis x right parenthesis equals fraction numerator 3 x squared minus 6 x minus 27 over denominator x minus 1 end fraction, maka left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis adalah:

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell row blank equals cell f left parenthesis x plus 1 right parenthesis end cell row blank equals cell fraction numerator 3 left parenthesis x plus 1 right parenthesis squared minus 6 left parenthesis x plus 1 right parenthesis minus 27 over denominator x minus 1 end fraction end cell row blank equals cell fraction numerator 3 left parenthesis x squared plus 2 x plus 1 right parenthesis minus 6 x minus 6 minus 27 over denominator x minus 1 end fraction end cell row blank equals cell fraction numerator 3 x squared plus 6 x plus 3 minus 6 x minus 6 minus 27 over denominator x minus 1 end fraction end cell row blank equals cell fraction numerator 3 x squared minus 30 over denominator x minus 1 end fraction end cell end table

h equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 3 x squared minus 30 over denominator x minus 1 end fraction end cell end table dan h(a)=9, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis a right parenthesis end cell equals 9 row cell fraction numerator 3 a squared minus 30 over denominator a minus 1 end fraction end cell equals 9 row cell 3 a squared minus 30 end cell equals cell 9 left parenthesis a minus 1 right parenthesis end cell row cell 3 a squared minus 30 end cell equals cell 9 a minus 9 end cell row cell 3 a squared minus 9 a minus 30 plus 9 end cell equals 0 row cell 3 a squared minus 9 a minus 21 end cell equals 0 end table

untuk mendapatkan nilai a gunakan rumus a equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction sehingga di peroleh:

table attributes columnalign right center left columnspacing 0px end attributes row a equals cell fraction numerator negative left parenthesis negative 9 right parenthesis plus-or-minus square root of left parenthesis negative 9 right parenthesis squared minus 4 times 3 times left parenthesis negative 21 right parenthesis end root over denominator 2 times 3 end fraction end cell row blank equals cell fraction numerator 9 plus-or-minus square root of 81 plus 251 end root over denominator 6 end fraction end cell row blank equals cell fraction numerator 9 plus-or-minus square root of 333 over denominator 6 end fraction end cell row blank equals cell fraction numerator 9 plus-or-minus square root of 9 times 37 end root over denominator 6 end fraction end cell row blank equals cell fraction numerator 9 plus-or-minus 3 square root of 37 over denominator 6 end fraction end cell end table

Dengan demikian, nilai a adalah fraction numerator 9 plus 3 square root of 37 over denominator 6 end fraction atau fraction numerator 9 minus 3 square root of 37 over denominator 6 end fraction.

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