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Diketahui fungsi f(x)=x2−1, g(x)=x−1x+1​, dan (f∘g∘h)(x)=x24x+4​. Rumus h(x)=...

Pertanyaan

Diketahui fungsi f left parenthesis x right parenthesis equals x squared minus 1g left parenthesis x right parenthesis equals fraction numerator x plus 1 over denominator x minus 1 end fraction, dan open parentheses f ring operator g ring operator h close parentheses left parenthesis x right parenthesis equals fraction numerator 4 x plus 4 over denominator x squared end fraction. Rumus  

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I. Kumaralalita

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

Diketahui fungsi berikut :

begin mathsize 14px style f open parentheses x close parentheses equals x squared minus 1 space g open parentheses x close parentheses equals fraction numerator x plus 1 over denominator x minus 1 end fraction space open parentheses f ring operator g ring operator h close parentheses open parentheses x close parentheses equals fraction numerator 4 x plus 4 over denominator x squared end fraction space end style 

Ditanya rumus dari h left parenthesis x right parenthesis.

Jadi, diperoleh hasil perhitungan sebagai berikut

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

Substitusikan pada komposisi dari tiga fungsi open parentheses f ring operator g ring operator h close parentheses left parenthesis x right parenthesis.

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

Perhatikan persamaan yang sebanding pada pembilang dan penyebut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 4 open parentheses h open parentheses x close parentheses close parentheses end cell equals cell 4 x plus 4 space end cell row cell h open parentheses x close parentheses end cell equals cell fraction numerator 4 x plus 4 over denominator 4 end fraction end cell row cell h open parentheses x close parentheses end cell equals cell x plus 1 end cell row blank blank blank row blank blank blank end table end style 

atau

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses h open parentheses x close parentheses minus 1 close parentheses squared end cell equals cell x squared end cell row cell h open parentheses x close parentheses minus 1 end cell equals x row cell h open parentheses x close parentheses end cell equals cell x plus 1 end cell end table end style 

Dengan demikian rumus dari h left parenthesis x right parenthesis equals x plus 1.

Jadi, jawaban yang tepat adalah C.

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