Jika f(x)=2x+2, g(x)=3−x, dan h(x)=4x−1, maka (g∘f∘h)−1(x) yang tepat adalah ....

Pertanyaan

Jika f left parenthesis x right parenthesis equals 2 x plus 2 comma space g left parenthesis x right parenthesis equals 3 minus x comma dan begin mathsize 14px style h left parenthesis x right parenthesis equals 4 x minus 1 end style, maka left parenthesis g ring operator f ring operator h right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis yang tepat adalah ....

  1. open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals 8 x plus 3 

  2. open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals 3 minus 8 x 

  3. open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus 3 over denominator 8 end fraction 

  4. open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 3 plus x over denominator 8 end fraction 

  5. open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 3 minus x over denominator 8 end fraction 

R. Setiawan

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Cara 1

Tentukan begin mathsize 14px style open parentheses g ring operator f ring operator h close parentheses open parentheses x close parentheses end style terlebih dahulu.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f ring operator h close parentheses open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses h open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell g open parentheses f open parentheses 4 x minus 1 close parentheses close parentheses end cell row blank equals cell g open parentheses 2 open parentheses 4 x minus 1 close parentheses plus 2 close parentheses end cell row blank equals cell g open parentheses 8 x minus 2 plus 2 close parentheses end cell row blank equals cell g open parentheses 8 x close parentheses end cell row blank equals cell 3 minus 8 x end cell end table end style 

 

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

Jika open parentheses g ring operator f ring operator h close parentheses open parentheses x close parentheses equals y, maka didapat perhitungan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f ring operator h right parenthesis left parenthesis x right parenthesis end cell equals cell 3 minus 8 x end cell row y equals cell 3 minus 8 x end cell row cell 8 x end cell equals cell 3 minus y end cell row x equals cell fraction numerator 3 minus y over denominator 8 end fraction end cell row cell left parenthesis g ring operator f ring operator h right parenthesis to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell fraction numerator 3 minus y over denominator 8 end fraction end cell end table end style 

Dengan demikian, begin mathsize 14px style open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 3 minus x over denominator 8 end fraction end cell end table end style.

 

Cara 2

Kita cari masing-masing begin mathsize 14px style f to the power of negative 1 end exponent comma space g to the power of negative 1 end exponent comma end style dan begin mathsize 14px style h to the power of negative 1 end exponent end style terlebih dahulu.

Dimulai dari mencari f to the power of negative 1 end exponent. Perhatikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell 2 x plus 2 end cell row y equals cell 2 x plus 2 end cell row x equals cell fraction numerator y minus 2 over denominator 2 end fraction end cell row cell f open parentheses x close parentheses end cell equals y row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals x row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator y minus 2 over denominator 2 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator x minus 2 over denominator 2 end fraction end cell end table

Kemudian, akan dicari g to the power of negative 1 end exponent. Perhatikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses x close parentheses end cell equals cell 3 minus x end cell row y equals cell 3 minus x end cell row x equals cell 3 minus y end cell row blank blank blank row cell g open parentheses x close parentheses end cell equals y row cell g to the power of negative 1 end exponent open parentheses y close parentheses end cell equals x row cell g to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell 3 minus y end cell row cell g to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell 3 minus x end cell end table

Selanjutnya, akan dicari h to the power of negative 1 end exponent. Perhatikan bahwa

table attributes columnalign right center left columnspacing 0px end attributes row cell h open parentheses x close parentheses end cell equals cell 4 x minus 1 end cell row y equals cell 4 x minus 1 end cell row x equals cell fraction numerator y plus 1 over denominator 4 end fraction end cell row blank blank blank row cell h open parentheses x close parentheses end cell equals y row cell h to the power of negative 1 end exponent open parentheses y close parentheses end cell equals x row cell h to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator y plus 1 over denominator 4 end fraction end cell row cell h to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator x plus 1 over denominator 4 end fraction end cell end table 

Ingat kembali bahwa begin mathsize 14px style open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses h to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end style, maka didapat hasil sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses h to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell h to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell h to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses 3 minus x close parentheses close parentheses end cell row blank equals cell h to the power of negative 1 end exponent open parentheses fraction numerator open parentheses 3 minus x close parentheses minus 2 over denominator 2 end fraction close parentheses end cell row blank equals cell h to the power of negative 1 end exponent open parentheses fraction numerator 1 minus x over denominator 2 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses fraction numerator 1 minus x over denominator 2 end fraction close parentheses plus 1 over denominator 4 end fraction end cell row blank equals cell fraction numerator fraction numerator 1 minus x over denominator 2 end fraction plus 2 over 2 over denominator 4 end fraction end cell row blank equals cell fraction numerator fraction numerator 3 minus x over denominator 2 end fraction over denominator 4 end fraction end cell row blank equals cell fraction numerator 3 minus x over denominator 8 end fraction end cell end table end style 

Jadi, jawaban yang tepat adalah E.

24

5.0 (2 rating)

M Dzakwan Hidayat

Pembahasan terpotong Jawaban tidak sesuai

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