Pertanyaan

Diketahui ( f − 1 ∘ g − 1 ∘ h − 1 ) ( x ) = 2 x − 4 ( h ∘ g ) ( x ) = 2 x + 1 x − 10 . Tentukan nilai 2 ( f ( 4 ) ) − 1 .

Diketahui

$(f^{- 1} \circ g^{- 1} \circ h^{- 1}) (x) = 2 x - 4$

$(h \circ g) (x) = \frac{x - 10}{2 x + 1}$ . Tentukan nilai $2 (f (4)) - 1$ .

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Pembahasan

Tentukan terlebih dahulu fungsi komposisi tiga fungsi. Ingatlah bahwa: Sehingga diperoleh: Kemudian carilahfungsi seperti berikut: Kemudian tentukan nilai . Maka, nilai dari adalah .

Tentukan terlebih dahulu fungsi komposisi tiga fungsi.

Ingatlah bahwa:

Sehingga diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell equals cell 2 x minus 4 end cell row cell open parentheses h ring operator g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell 2 x minus 4 end cell row y equals cell 2 x minus 4 end cell row cell 2 x minus 4 end cell equals y row cell 2 x end cell equals cell y plus 4 end cell row x equals cell fraction numerator y plus 4 over denominator 2 end fraction end cell row cell open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses end cell equals cell fraction numerator x plus 4 over denominator 2 end fraction end cell row cell open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses end cell equals cell 1 half x plus 2 end cell end table$

Kemudian carilah fungsi f open parentheses x close parentheses seperti berikut:

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses h ring operator g ring operator f close parentheses open parentheses x close parentheses end cell equals cell 1 half x plus 2 end cell row cell open parentheses h ring operator g close parentheses open parentheses f open parentheses x close parentheses close parentheses end cell equals cell 1 half x plus 2 end cell row cell fraction numerator f open parentheses x close parentheses minus 10 over denominator 2 times f open parentheses x close parentheses plus 1 end fraction end cell equals cell 1 half x plus 2 end cell row cell f open parentheses x close parentheses minus 10 end cell equals cell open parentheses 2 times f open parentheses x close parentheses plus 1 close parentheses open parentheses 1 half x plus 2 close parentheses end cell row cell f open parentheses x close parentheses minus 10 end cell equals cell x times f open parentheses x close parentheses plus 4 times f open parentheses x close parentheses plus 1 half x plus 2 end cell row cell f open parentheses x close parentheses minus x times f open parentheses x close parentheses minus 4 times f open parentheses x close parentheses end cell equals cell 1 half x plus 10 plus 2 end cell row cell negative 3 times f open parentheses x close parentheses minus x times f open parentheses x close parentheses end cell equals cell 1 half x plus 12 end cell row cell f open parentheses x close parentheses times open parentheses negative 3 minus x close parentheses end cell equals cell 1 half x plus 12 end cell row cell f open parentheses x close parentheses end cell equals cell fraction numerator 1 half x plus 12 over denominator negative 3 minus x end fraction end cell end table$

Kemudian tentukan nilai 2 open parentheses f open parentheses 4 close parentheses close parentheses minus 1 .

$table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell fraction numerator 1 half x plus 12 over denominator negative 3 minus x end fraction end cell row cell 2 open parentheses f open parentheses 4 close parentheses close parentheses minus 1 end cell equals cell 2 open parentheses fraction numerator begin display style 1 half end style open parentheses 4 close parentheses plus 12 over denominator negative 3 minus 4 end fraction close parentheses minus 1 end cell row blank equals cell 2 open parentheses fraction numerator 2 plus 12 over denominator negative 7 end fraction close parentheses minus 1 end cell row blank equals cell 2 open parentheses fraction numerator 14 over denominator negative 7 end fraction close parentheses minus 1 end cell row blank equals cell 2 open parentheses negative 2 close parentheses minus 1 end cell row blank equals cell negative 4 minus 1 end cell row blank equals 5 end table$

Maka, nilai dari 2 open parentheses f open parentheses 4 close parentheses close parentheses minus 1 adalah .