Roboguru

Ditentukan fungsi f:R→R, g:R→R, dan h:R→R dengan f(x)=x+41​, g(x)=3x, dan h(x)=x−1. Rumus (h∘g∘f)−1(1−x)=....

Pertanyaan

Ditentukan fungsi begin mathsize 14px style f colon space straight R rightwards arrow straight R end styleError converting from MathML to accessible text., dan Error converting from MathML to accessible text. dengan begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 4 end fraction end stylebegin mathsize 14px style g open parentheses x close parentheses equals 3 x end style, dan begin mathsize 14px style h open parentheses x close parentheses equals x minus 1 end style. Rumus begin mathsize 14px style open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses 1 minus x close parentheses equals... end style

  1. begin mathsize 14px style fraction numerator 4 x minus 5 over denominator x minus 2 end fraction semicolon space x not equal to 2 end style 

  2. begin mathsize 14px style fraction numerator 4 x minus 5 over denominator 2 minus x end fraction semicolon space x not equal to 2 end style 

  3. begin mathsize 14px style fraction numerator 4 x plus 5 over denominator x plus 2 end fraction semicolon space x not equal to negative 2 end style 

  4. begin mathsize 14px style fraction numerator 4 x plus 5 over denominator 2 minus x end fraction semicolon space x not equal to 2 end style 

  5. begin mathsize 14px style fraction numerator 4 x minus 5 over denominator x plus 2 end fraction semicolon space x not equal to negative 2 end style  

Pembahasan Soal:

Gunakan konsep komposisi fungsi dan invers fungsi.

Diketahui:

begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 4 end fraction end style

begin mathsize 14px style g open parentheses x close parentheses equals 3 x end style

begin mathsize 14px style h open parentheses x close parentheses equals x minus 1 end style

Akan ditentukan begin mathsize 14px style left parenthesis h ring operator g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis 1 minus x right parenthesis end style.

Terlebih dahulu tentukan begin mathsize 14px style left parenthesis h ring operator g ring operator f right parenthesis open parentheses x close parentheses end style, diperoleh:

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

Kemudian tentukan begin mathsize 14px style left parenthesis h ring operator g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end style, perhatikan perhitungan berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell left parenthesis h ring operator g ring operator f right parenthesis left parenthesis x right parenthesis equals y end cell equals cell fraction numerator negative x minus 1 over denominator x plus 4 end fraction end cell row cell y open parentheses x plus 4 close parentheses end cell equals cell negative x minus 1 end cell row cell x y plus 4 y end cell equals cell negative x minus 1 end cell row cell x y plus x end cell equals cell negative 4 y minus 1 end cell row cell x open parentheses y plus 1 close parentheses end cell equals cell negative 4 y minus 1 end cell row x equals cell fraction numerator negative 4 y minus 1 over denominator y plus 1 end fraction end cell row cell left parenthesis h ring operator g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator negative 4 x minus 1 over denominator x plus 1 end fraction end cell end table end style 

Sehingga begin mathsize 14px style left parenthesis h ring operator g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis 1 minus x right parenthesis end style dapat diperoleh sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell left parenthesis h ring operator g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator negative 4 x minus 1 over denominator x plus 1 end fraction end cell row cell left parenthesis h ring operator g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis 1 minus x right parenthesis end cell equals cell fraction numerator negative 4 open parentheses 1 minus x close parentheses minus 1 over denominator open parentheses 1 minus x close parentheses plus 1 end fraction end cell row blank equals cell fraction numerator negative 4 plus 4 x minus 1 over denominator 1 minus x plus 1 end fraction end cell row blank equals cell fraction numerator 4 x minus 5 over denominator negative x plus 2 end fraction end cell row cell left parenthesis h ring operator g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis 1 minus x right parenthesis end cell equals cell fraction numerator 4 x minus 5 over denominator 2 minus x end fraction space semicolon space x not equal to 2 end cell end table end style 

Jadi, jawaban yang tepat adalah B.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

Y. Fathoni

Mahasiswa/Alumni Universitas Negeri Yogyakarta.

Terakhir diupdate 05 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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