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Diketahui fungsi f ( x ) = x 3 dan g ( x ) = 3 x − 4 . Jika a = ( g − 1 ∘ f − 1 ) ( 8 ) , maka nilai ( f − 1 ∘ g − 1 ) ( 10 a ) adalah ...

Diketahui fungsi  dan . Jika , maka nilai  adalah ...

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S. Nur

Master Teacher

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Pembahasan

Tentukan invers fungsinya terlebih dahulu. Kemudian dari fungsi-fungsi invers tersebut gunakan rumus fungsi komposisi untuk mengetahui nilai . Kemudian diperoleh: Maka, nilai adalah .

Tentukan invers fungsinya terlebih dahulu.

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell x cubed end cell row y equals cell x cubed end cell row x equals cell cube root of y end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell cube root of x end cell row blank blank blank row cell g open parentheses x close parentheses end cell equals cell 3 x minus 4 end cell row y equals cell 3 x minus 4 end cell row x equals cell fraction numerator y plus 4 over denominator 3 end fraction end cell row cell g to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator x plus 4 over denominator 3 end fraction end cell end table

Kemudian dari fungsi-fungsi invers tersebut gunakan rumus fungsi komposisi untuk mengetahui nilai a.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses 8 close parentheses end cell equals a row cell g to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses 8 close parentheses close parentheses end cell equals a row cell g to the power of negative 1 end exponent open parentheses cube root of 8 close parentheses end cell equals a row cell fraction numerator cube root of 8 plus 4 over denominator 3 end fraction end cell equals a row cell fraction numerator 2 plus 4 over denominator 3 end fraction end cell equals a row cell 6 over 3 end cell equals a row 2 equals a end table

Kemudian diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses 10 a close parentheses end cell equals cell f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent open parentheses 10 open parentheses 2 close parentheses close parentheses close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent open parentheses 20 close parentheses close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses fraction numerator 20 plus 4 over denominator 3 end fraction close parentheses end cell row blank equals cell cube root of fraction numerator 20 plus 4 over denominator 3 end fraction end root end cell row blank equals cell cube root of 24 over 3 end root end cell row blank equals cell cube root of 8 end cell row blank equals 2 end table

Maka, nilai open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses 10 a close parentheses adalah 2.

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