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 $f:P\rightarrow N$ ; $g:N\rightarrow R$ dan $h:P\rightarrow R$ dengan $g(x)=x+1$ , $f left parenthesis x right parenthesis equals fraction numerator 3 x squared minus 6 x minus 27 over denominator x minus 1 end fraction$ dan $h=f\circ g$ . Tentukanlah nilai $a$ agar $h(a)=9$ .

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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 $f:y\rightarrow z$ dan $g:x\rightarrow y$ . Komposisi fungsi $g$ dan $f$ yang memetakan $x$ ke $y$ kemudian ke $z$ adalah:

$\begin{array}{rcl}{(f\circ g)(x)}&=&{f(g(x))}\end{array}$

Misalkan terdapat persamaan kuadrat $ax^2+bx+c=0$ , 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 $g(x)=x+1$ 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 $(f\circ g)(x)$ 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 $fraction numerator 9 minus 3 square root of 37 over denominator 6 end fraction$ .