Jika f(x)=x−3, g(x)=x−12​ dan h(x)=4+x2​. Tunjukkan bahwa (h∘g∘f)(x)=(h∘g∘f)−1(x).

Pertanyaan

Jika f open parentheses x close parentheses equals x minus 3 comma space g open parentheses x close parentheses equals fraction numerator 2 over denominator x minus 1 end fraction dan h open parentheses x close parentheses equals 4 plus 2 over x. Tunjukkan bahwa open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses equals open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses.

A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Jawaban

nilai bold left parenthesis bold italic h bold ring operator bold italic g bold ring operator bold italic f bold right parenthesis bold left parenthesis bold italic x bold right parenthesis bold equals bold left parenthesis bold italic h bold ring operator bold italic g bold ring operator bold italic f bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold italic x bold right parenthesis adalah sama-sama bold italic x.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut bold left parenthesis bold italic h bold ring operator bold italic g bold ring operator bold italic f bold right parenthesis bold left parenthesis bold italic x bold right parenthesis bold equals bold left parenthesis bold italic h bold ring operator bold italic g bold ring operator bold italic f bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold italic x bold right parenthesis adalah sama-sama bold italic x.

Ingat!

  1. Misalkan fungsi f colon A rightwards arrow B, fungsi g colon B rightwards arrow C, dan fungsi h colon C rightwards arrow D, maka terdapat komposisi dari tiga fungsi. yaitu open parentheses h ring operator g ring operator f close parentheses colon A rightwards arrow D.
  2. open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses equals h open parentheses g open parentheses f open parentheses x close parentheses close parentheses close parentheses

Diketahui:

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell x minus 3 end cell row cell g open parentheses x close parentheses end cell equals cell fraction numerator 2 over denominator x minus 1 end fraction end cell row cell h open parentheses x close parentheses end cell equals cell 4 plus 2 over x end cell end table

Ditanya: Tunjukkan bahwa open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses equals open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses.

Jawab: 

Dari rumus open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses equals h open parentheses g open parentheses f open parentheses x close parentheses close parentheses close parentheses, maka harus mencari g open parentheses f open parentheses x close parentheses close parentheses dahulu.

table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses f open parentheses x close parentheses close parentheses end cell equals cell g open parentheses x minus 3 close parentheses end cell row blank equals cell fraction numerator 2 over denominator open parentheses x minus 3 close parentheses minus 1 end fraction end cell row blank equals cell fraction numerator 2 over denominator x minus 4 end fraction end cell end table

Mencari open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses equals h open parentheses g open parentheses f open parentheses x close parentheses close parentheses close parentheses:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses end cell equals cell h open parentheses g open parentheses f open parentheses x close parentheses close parentheses close parentheses end cell row blank equals cell h open parentheses fraction numerator 2 over denominator x minus 4 end fraction close parentheses end cell row blank equals cell 4 plus fraction numerator 2 over denominator begin display style fraction numerator 2 over denominator x minus 4 end fraction end style end fraction end cell row blank equals cell 4 plus up diagonal strike 2 times fraction numerator x minus 4 over denominator up diagonal strike 2 end fraction end cell row blank equals cell 4 plus x minus 4 end cell row blank equals x end table

Mencari table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell end table:

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses equals x end cell row y equals x row x equals y row cell open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses y close parentheses end cell equals y row cell open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals x end table

Dengan demikian, nilai bold left parenthesis bold italic h bold ring operator bold italic g bold ring operator bold italic f bold right parenthesis bold left parenthesis bold italic x bold right parenthesis bold equals bold left parenthesis bold italic h bold ring operator bold italic g bold ring operator bold italic f bold right parenthesis to the power of bold minus bold 1 end exponent bold left parenthesis bold italic x bold right parenthesis adalah sama-sama bold italic x.

66

4.2 (7 rating)

Pertanyaan serupa

Jika f(x)=x+2, g(x)=x1​ dan h(x−1)=x−3. Tentukanlah (h∘g∘f)−1(x).

83

4.1

Jawaban terverifikasi

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