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 $(f\circ g)(x)=(g\circ f)(x)$ .

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

terbukti $(f\circ g)(x)=(g\circ f)(x)$ .

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah terbukti $(f\circ g)(x)=(g\circ f)(x)$ .

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

$(g\circ f)(x)=g(f(x))$

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 $(f\circ g)(x)=(g\circ f)(x)$ .

Pertama akan dicari $(f\circ g)(x)$ .

$table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell f left parenthesis g left parenthesis x right parenthesis end cell row blank equals cell f open parentheses fraction numerator 5 x plus 3 over denominator 4 x minus 2 end fraction close parentheses end cell row blank equals cell fraction numerator 2 open parentheses fraction numerator 5 x plus 3 over denominator 4 x minus 2 end fraction close parentheses plus 3 over denominator 4 open parentheses fraction numerator 5 x plus 3 over denominator 4 x minus 2 end fraction close parentheses minus 5 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 10 x plus 6 over denominator 4 x minus 2 end fraction end style plus 3 over denominator begin display style fraction numerator 20 x plus 12 over denominator 4 x minus 2 end fraction end style minus 5 end fraction end cell end table$

dikali dengan $4x-2$ 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 $(g\circ f)(x)$ .

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

dikali dengan $4x-5$ 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 $(f\circ g)(x)=(g\circ f)(x)$ .