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Misal f(x)=4x−52x+3​ dan g(x)=4x−25x+3​  tunjukkan bahwa (f∘g)(x)=(g∘f)(x) .

Pertanyaan

Misal f left parenthesis x right parenthesis equals fraction numerator 2 x plus 3 over denominator 4 x minus 5 end fraction dan g left parenthesis x right parenthesis equals fraction numerator 5 x plus 3 over denominator 4 x minus 2 end fraction  tunjukkan bahwa left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis .

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Master Teacher

Jawaban terverifikasi

Jawaban

terbukti left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah terbukti left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis.

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

left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis equals g left parenthesis f left parenthesis x right parenthesis right parenthesis

Diketahui  f left parenthesis x right parenthesis equals fraction numerator 2 x plus 3 over denominator 4 x minus 5 end fraction dan g left parenthesis x right parenthesis equals fraction numerator 5 x plus 3 over denominator 4 x minus 2 end fraction. Akan dibuktikan left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis

Pertama akan dicari left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis.

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 end cell row blank equals cell f open parentheses fraction numerator 5 x plus 3 over denominator 4 x minus 2 end fraction close parentheses end cell row blank equals cell fraction numerator 2 open parentheses fraction numerator 5 x plus 3 over denominator 4 x minus 2 end fraction close parentheses plus 3 over denominator 4 open parentheses fraction numerator 5 x plus 3 over denominator 4 x minus 2 end fraction close parentheses minus 5 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 10 x plus 6 over denominator 4 x minus 2 end fraction end style plus 3 over denominator begin display style fraction numerator 20 x plus 12 over denominator 4 x minus 2 end fraction end style minus 5 end fraction end cell end table

dikali dengan 4 x minus 2 sehingga diperoleh:

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 fraction numerator 10 x plus 6 plus 3 left parenthesis 4 x minus 2 right parenthesis over denominator 20 x plus 12 minus 5 left parenthesis 4 x minus 2 right parenthesis end fraction end cell row blank equals cell fraction numerator 10 x plus 6 plus 12 x minus 6 over denominator 20 x plus 12 minus 20 x plus 10 end fraction end cell row blank equals cell fraction numerator 22 x over denominator 22 end fraction end cell row blank equals x end table

Selanjutnya akan dicari left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis.

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals cell g left parenthesis f left parenthesis x right parenthesis right parenthesis end cell row blank equals cell g open parentheses fraction numerator 2 x plus 3 over denominator 4 x minus 5 end fraction close parentheses end cell row blank equals cell fraction numerator 5 open parentheses fraction numerator 2 x plus 3 over denominator 4 x minus 5 end fraction close parentheses plus 3 over denominator 4 open parentheses fraction numerator 2 x plus 3 over denominator 4 x minus 5 end fraction close parentheses minus 2 end fraction end cell row blank equals cell fraction numerator fraction numerator 10 x plus 15 over denominator 4 x minus 5 end fraction plus 3 over denominator fraction numerator 8 x plus 12 over denominator 4 x minus 5 end fraction minus 2 end fraction end cell end table

dikali dengan 4 x minus 5 sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator 10 x plus 15 plus 3 left parenthesis 4 x minus 5 right parenthesis over denominator 8 x plus 12 minus 2 left parenthesis 4 x minus 5 right parenthesis end fraction end cell row blank equals cell fraction numerator 10 x plus 15 plus 12 x minus 15 over denominator 8 x plus 12 minus 8 x plus 10 end fraction end cell row blank equals cell fraction numerator 22 x over denominator 22 end fraction end cell row blank equals x end table

Dengan demikian, terbukti left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis.

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