Diketahui f(x)=x+32x,x=−3 dan g(x)=xx+1,x=0. Fungsi (f∘g)−1(31) adalah...

Pertanyaan

Diketahui $begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 2 x over denominator x plus 3 end fraction comma x not equal to negative 3 end style$ dan $begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator x plus 1 over denominator x end fraction comma x not equal to 0 end style$ . Fungsi $\style{font-size:14px}{\left(f\circ g\right)^{-1}\left(\frac13\right)}$ adalah...

S. Amamah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

Dengan menggunakan konsep invers fungsi komposisi maka diperoleh

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell f open parentheses fraction numerator x plus 1 over denominator x end fraction close parentheses end cell row blank equals cell fraction numerator 2 open parentheses begin display style fraction numerator x plus 1 over denominator x end fraction end style close parentheses over denominator begin display style fraction numerator x plus 1 over denominator x end fraction end style plus 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 x plus 2 over denominator x end fraction end style over denominator begin display style fraction numerator x plus 1 over denominator x end fraction end style plus begin display style fraction numerator 3 x over denominator x end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 2 x plus 2 over denominator x end fraction end style over denominator begin display style fraction numerator 4 x plus 1 over denominator x end fraction end style end fraction end cell row blank equals cell fraction numerator 2 x plus 2 over denominator 4 x plus 1 end fraction end cell end table end style$

Misalkan $\left(f\circ g\right)\left(x\right)=y$ maka invers dari $\left(f\circ g\right)\left(x\right)$ adalah $\left(f\circ g\right)^{-1}\left(y\right)=x$ :

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator 2 x plus 2 over denominator 4 x plus 1 end fraction end cell row y equals cell fraction numerator 2 x plus 2 over denominator 4 x plus 1 end fraction end cell row cell y open parentheses 4 x plus 1 close parentheses end cell equals cell 2 x plus 2 end cell row cell 4 x y minus 2 x end cell equals cell 2 minus y end cell row cell x open parentheses 4 x minus 2 close parentheses end cell equals cell 2 minus y end cell row x equals cell fraction numerator 2 minus y over denominator 4 x minus 2 end fraction end cell end table end style$

Sehingga diperoleh

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 2 minus x over denominator 4 x minus 2 end fraction end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 1 third close parentheses end cell equals cell fraction numerator 2 minus begin display style 1 third end style over denominator 4 open parentheses begin display style 1 third end style close parentheses minus 2 end fraction end cell row blank equals cell fraction numerator begin display style 5 over 3 end style over denominator negative begin display style 2 over 3 end style end fraction end cell row blank equals cell negative 5 over 2 end cell end table end style$

Dengnan demikian nilai dari .

Oleh karena itu, jawaban yang benar adalah B.