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Diketahui f ( x + 2 ) = 4 x − 9 dan ( g ∘ f ) ( x ) = 4 x − 16 1 ​ , x  = 4 . Tentukan: a. rumus fungsi f b. rumus fungsi g c. rumus ( f ∘ g ) ( x )

Diketahui  dan . Tentukan:

a. rumus fungsi 

b. rumus fungsi  

c. rumus  

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S. Amamah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

rumus fungsi adalah .

rumus fungsi open parentheses f ring operator g close parentheses open parentheses x close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f ring operator g close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 17 x minus 13 over denominator x plus 1 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank not equal to blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table.

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Pembahasan

a. Misalkan maka diperoleh kemudian substitusikan maka diperoleh: Jadirumus fungsi adalah . b. Ingat konsep komposisi fungsi: maka: Misalkan maka diperoleh sehingga: Jadi, rumus fungsi adalah . c. dan maka : Dengan demikian rumus fungsi adalah .

a. Misalkan y equals x plus 2 maka diperoleh x equals y minus 2 kemudian substitusikan maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x plus 2 close parentheses end cell equals cell 4 x minus 9 end cell row cell f open parentheses y close parentheses end cell equals cell 4 open parentheses y minus 2 close parentheses minus 9 end cell row blank equals cell 4 y minus 8 minus 9 end cell row blank equals cell 4 y minus 17 end cell row cell f open parentheses y close parentheses end cell equals cell 4 y minus 17 end cell row cell f open parentheses x close parentheses end cell equals cell 4 x minus 17 end cell end table 

Jadi rumus fungsi f adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank f end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 17 end table

b. Ingat konsep komposisi fungsi:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses x close parentheses close parentheses end cell row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses g open parentheses x close parentheses close parentheses end cell end table

maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell fraction numerator 1 over denominator 4 x minus 16 end fraction end cell row cell g open parentheses f open parentheses x close parentheses close parentheses end cell equals cell fraction numerator 1 over denominator 4 x minus 16 end fraction end cell row cell g open parentheses 4 x minus 17 close parentheses end cell equals cell fraction numerator 1 over denominator 4 x minus 16 end fraction end cell end table

Misalkan y equals 4 x minus 17 maka diperoleh x equals fraction numerator y plus 17 over denominator 4 end fraction sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses 4 x minus 17 close parentheses end cell equals cell fraction numerator 1 over denominator 4 x minus 16 end fraction end cell row cell g open parentheses y close parentheses end cell equals cell fraction numerator 1 over denominator 4 open parentheses begin display style fraction numerator y plus 17 over denominator 4 end fraction end style close parentheses minus 16 end fraction end cell row blank equals cell fraction numerator 1 over denominator y plus 17 minus 16 end fraction end cell row cell g open parentheses y close parentheses end cell equals cell fraction numerator 1 over denominator y plus 1 end fraction end cell row cell g open parentheses x close parentheses end cell equals cell fraction numerator 1 over denominator x plus 1 end fraction end cell end table 

Jadi, rumus fungsi g adalah g open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 1 end fraction

c. table attributes columnalign right center left columnspacing 0px end attributes row blank blank f end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 17 end table dan g open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 1 end fraction maka open parentheses f ring operator g close parentheses open parentheses x close parentheses:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell f open parentheses fraction numerator 1 over denominator x plus 1 end fraction close parentheses end cell row blank equals cell 4 open parentheses fraction numerator 1 over denominator x plus 1 end fraction close parentheses minus 17 end cell row blank equals cell fraction numerator 4 over denominator x plus 1 end fraction minus fraction numerator 17 open parentheses x plus 1 close parentheses over denominator x plus 1 end fraction end cell row blank equals cell fraction numerator 4 minus 17 x minus 17 over denominator x plus 1 end fraction end cell row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator negative 17 x minus 13 over denominator x plus 1 end fraction comma space x not equal to negative 1 end cell end table 

Dengan demikian rumus fungsi open parentheses f ring operator g close parentheses open parentheses x close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f ring operator g close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 17 x minus 13 over denominator x plus 1 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank not equal to blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table.

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