Fungsi f : R→R dan g : R→R ditentukan oleh rumus f(x)=7x−5 dan g(x)=4x+9. Tentukan Apakah (g∘f)−1(x)= (f−1∘g−1)(x) (Ya / Tidak)

Pertanyaan

Fungsi $\style{font-size:14px}{\mathrm f\;:\;\mathrm R\rightarrow\mathrm R}$ dan $\style{font-size:14px}{\mathrm g\;:\;\mathrm R\rightarrow\mathrm R}$ ditentukan oleh rumus $\style{font-size:14px}{\mathrm f\left(\mathrm x\right)=7\mathrm x-5}$ dan $\style{font-size:14px}{\mathrm g\left(\mathrm x\right)=4\mathrm x+9}$ . Tentukan

Apakah $\style{font-size:14px}{\left(\mathrm g\circ\mathrm f\right)^{-1}\left(\mathrm x\right)=}$ $\style{font-size:14px}{\left(\mathrm f^{-1}\circ\mathrm g^{-1}\right)\left(\mathrm x\right)}$ (Ya / Tidak)

R. Novianto

Master Teacher

Mahasiswa/Alumni Universitas Tanjungpura Pontianak

Jawaban terverifikasi

Jawaban

ya terbukti bahwa

Pembahasan

Pertama kita tentukan invers dari

$begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell straight f open parentheses straight x close parentheses end cell equals cell 7 straight x minus 5 end cell row straight y equals cell 7 straight x minus 5 end cell row cell negative 7 straight x end cell equals cell negative straight y minus 5 end cell row straight x equals cell fraction numerator negative straight y minus 5 over denominator negative 7 end fraction end cell row sehingga colon blank row cell straight f to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell fraction numerator negative straight x minus 5 over denominator negative 7 end fraction end cell end table end style$

Selanjutnya kita tentukan invers dari

$begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell straight g open parentheses straight x close parentheses end cell equals cell 4 straight x plus 9 end cell row straight y equals cell 4 straight x plus 9 end cell row cell negative 4 straight x end cell equals cell negative straight y plus 9 end cell row straight x equals cell fraction numerator negative straight y plus 9 over denominator negative 4 end fraction end cell row sehingga colon blank row cell straight g to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell fraction numerator negative straight x plus 9 over denominator negative 4 end fraction end cell end table end style$

Maka sebagai berikut :

$begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses straight f to the power of negative 1 end exponent ring operator straight g to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses end cell equals cell fraction numerator negative open parentheses straight g to the power of negative 1 end exponent open parentheses straight x close parentheses close parentheses minus 5 over denominator negative 7 end fraction end cell row blank equals cell fraction numerator negative open parentheses begin display style fraction numerator negative straight x plus 9 over denominator negative 4 end fraction end style close parentheses minus 5 over denominator negative 7 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator straight x minus 9 over denominator negative 4 end fraction minus open parentheses fraction numerator negative 20 over denominator negative 4 end fraction close parentheses end style over denominator negative 7 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator straight x minus 9 plus 20 over denominator negative 4 end fraction end style over denominator negative 7 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator straight x plus 11 over denominator negative 4 end fraction end style over denominator negative 7 end fraction end cell row blank equals cell fraction numerator straight x plus 11 over denominator open parentheses negative 4 close parentheses times open parentheses negative 7 close parentheses end fraction end cell row blank equals cell fraction numerator straight x plus 11 over denominator 28 end fraction end cell end table end style$

maka diperoleh, $begin mathsize 14px style open parentheses straight f to the power of negative 1 end exponent ring operator straight g to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses equals fraction numerator straight x plus 11 over denominator 28 end fraction end style$

Kemudian kita tentukan begin mathsize 14px style open parentheses straight g ring operator straight f to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses end style

$begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses straight g ring operator straight f close parentheses open parentheses straight x close parentheses end cell equals cell 4 open parentheses straight f open parentheses straight x close parentheses close parentheses plus 9 end cell row blank equals cell 4 open parentheses 7 straight x minus 5 close parentheses plus 9 end cell row blank equals cell 28 straight x minus 20 plus 9 end cell row blank equals cell 28 straight x minus 11 end cell end table Dijadikan space invers table attributes columnalign right center left columnspacing 2px end attributes row straight y equals cell 28 straight x minus 11 end cell row cell negative 28 straight x end cell equals cell negative straight y minus 11 end cell row straight x equals cell fraction numerator negative straight y minus 11 over denominator negative 28 end fraction end cell row straight x equals cell fraction numerator straight y plus 11 over denominator 28 end fraction end cell row maka colon blank row cell open parentheses straight g ring operator straight f close parentheses to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell fraction numerator straight x plus 11 over denominator 28 end fraction end cell end table end style$

maka diperoleh, $begin mathsize 14px style open parentheses straight g ring operator straight f close parentheses to the power of negative 1 end exponent open parentheses straight x close parentheses equals fraction numerator straight x plus 11 over denominator 28 end fraction end style$ dan hasilnya sama dengan $begin mathsize 14px style open parentheses straight f to the power of negative 1 end exponent ring operator straight g to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses equals fraction numerator straight x plus 11 over denominator 28 end fraction end style$

Jadi, ya terbukti bahwa