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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 begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 1 third close parentheses end style adalah...

  1. begin mathsize 14px style negative 7 over 2 end style

  2. begin mathsize 14px style negative 5 over 2 end style

  3. undefined

  4. begin mathsize 14px style 5 over 2 end style

  5. begin mathsize 14px style 7 over 2 end style

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 open parentheses f ring operator g close parentheses open parentheses x close parentheses equals y maka invers dari open parentheses f ring operator g close parentheses open parentheses x close parentheses adalah open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses y close parentheses equals 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 begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 1 third close parentheses equals negative 5 over 2 end style.

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