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Diketahui f(x)=2x+3, g(x)=3x−1, dan (h∘g∘f)(x)=x+4x2−1​. Tentukan nilai h(0).

Pertanyaan

Diketahui f left parenthesis x right parenthesis equals 2 x plus 3g left parenthesis x right parenthesis equals 3 x minus 1, dan left parenthesis h ring operator g ring operator f right parenthesis left parenthesis x right parenthesis equals fraction numerator x squared minus 1 over denominator x plus 4 end fraction. Tentukan nilai h left parenthesis 0 right parenthesis.

 

A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Jawaban

didapat nilai dari h left parenthesis 0 right parenthesis equals 28 over 96 equals 7 over 24

Pembahasan

Ingat bahwa left parenthesis h ring operator g ring operator f right parenthesis left parenthesis x right parenthesis equals h left parenthesis g left parenthesis f left parenthesis x right parenthesis right parenthesis right parenthesis. Sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis h ring operator g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator x squared minus 1 over denominator x plus 4 end fraction end cell row cell h left parenthesis g left parenthesis f left parenthesis x right parenthesis right parenthesis right parenthesis end cell equals cell fraction numerator x squared minus 1 over denominator x plus 4 end fraction end cell row cell h left parenthesis g left parenthesis 2 x plus 3 right parenthesis end cell equals cell fraction numerator x squared minus 1 over denominator x plus 4 end fraction end cell row cell h left parenthesis 3 left parenthesis 2 x plus 3 right parenthesis minus 1 right parenthesis end cell equals cell fraction numerator x squared minus 1 over denominator x plus 4 end fraction end cell row cell h left parenthesis 6 x plus 9 minus 1 right parenthesis end cell equals cell fraction numerator x squared minus 1 over denominator x plus 4 end fraction end cell row cell h left parenthesis 6 x plus 8 right parenthesis end cell equals cell fraction numerator x squared minus 1 over denominator x plus 4 end fraction end cell end table

Misalkan: 6 x plus 8 equals y maka 

table attributes columnalign right center left columnspacing 0px end attributes row cell 6 x end cell equals cell y minus 8 end cell row x equals cell fraction numerator y minus 8 over denominator 6 end fraction end cell end table

Jadi, 

table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis 6 x plus 8 right parenthesis end cell equals cell fraction numerator x squared minus 1 over denominator x plus 4 end fraction end cell row cell h left parenthesis y right parenthesis end cell equals cell fraction numerator open parentheses begin display style fraction numerator y minus 8 over denominator 6 end fraction end style close parentheses squared minus 1 over denominator open parentheses begin display style fraction numerator y minus 8 over denominator 6 end fraction end style close parentheses plus 4 end fraction end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator y squared minus 16 y plus 64 over denominator 36 end fraction end style close parentheses minus 1 over denominator open parentheses begin display style fraction numerator y minus 8 over denominator 6 end fraction end style close parentheses plus 4 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator y squared minus 16 y plus 64 over denominator 36 end fraction end style minus begin display style 36 over 36 end style over denominator begin display style fraction numerator y minus 8 over denominator 6 end fraction end style plus begin display style 24 over 6 end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator y squared minus 16 y plus 28 over denominator 36 end fraction end style over denominator begin display style fraction numerator y plus 16 over denominator 6 end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator y squared minus 16 y plus 28 over denominator 36 end fraction end style over denominator begin display style fraction numerator 6 left parenthesis y plus 16 right parenthesis over denominator 36 end fraction end style end fraction end cell row cell h left parenthesis y right parenthesis end cell equals cell fraction numerator y squared minus 16 y plus 28 over denominator 6 left parenthesis y plus 16 right parenthesis end fraction end cell row cell h left parenthesis x right parenthesis end cell equals cell fraction numerator x squared minus 16 x plus 28 over denominator 6 left parenthesis x plus 16 right parenthesis end fraction end cell end table

Sehingga nilai dari h left parenthesis 0 right parenthesis adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis end cell equals cell fraction numerator x squared minus 16 x plus 28 over denominator 6 left parenthesis x plus 16 right parenthesis end fraction end cell row cell h left parenthesis 0 right parenthesis end cell equals cell fraction numerator left parenthesis 0 right parenthesis squared minus 16 left parenthesis 0 right parenthesis plus 28 over denominator 6 left parenthesis 0 plus 16 right parenthesis end fraction end cell row blank equals cell fraction numerator 0 minus 0 plus 28 over denominator 6 left parenthesis 16 right parenthesis end fraction end cell row blank equals cell 28 over 96 end cell end table

Dengan demikian, didapat nilai dari h left parenthesis 0 right parenthesis equals 28 over 96 equals 7 over 24

23

1.0 (1 rating)

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