Pertanyaan

Diketahui f ( x ) = x + 1 2 x − 1 , g ( x ) = x 2 − 1 , dan h ( x ) = 3 x + 5 . Tentukan: f. nilai jika ( f − 1 ∘ h − 1 ∘ g − 1 ) ( a ) = 2 ,

Diketahui $f (x) = \frac{2 x - 1}{x + 1}, g (x) = x^{2} - 1$ , dan $h (x) = 3 x + 5$ . Tentukan:

f. nilai $a$ jika $(f^{- 1} \circ h^{- 1} \circ g^{- 1}) (a) = 2$ ,

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W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

nilai adalah .

nilai

adalah

Pembahasan

Diketahui , dan . Maka: - Menentukan - Menentukan - Menentukan Sehingga: Kemudian: Jadi, nilai adalah .

Diketahui $f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 1 end fraction comma space g open parentheses x close parentheses equals x squared minus 1$ , dan h open parentheses x close parentheses equals 3 x plus 5 . Maka:

- Menentukan f to the power of negative 1 end exponent open parentheses x close parentheses

$space space space space space space f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 1 end fraction space space space space space space space space space space y equals fraction numerator 2 x minus 1 over denominator x plus 1 end fraction y open parentheses x plus 1 close parentheses equals 2 x minus 1 space space space x y plus y equals 2 x minus 1 space x y minus 2 x equals negative 1 minus y x open parentheses y minus 2 close parentheses equals negative 1 minus y space space space space space space space space space space x equals fraction numerator negative 1 minus y over denominator y minus 2 end fraction space space f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative 1 minus x over denominator x minus 2 end fraction$

- Menentukan g to the power of negative 1 end exponent open parentheses x close parentheses

- Menentukan h to the power of negative 1 end exponent open parentheses x close parentheses

$space space space space space h open parentheses x close parentheses equals 3 x plus 5 space space space space space space space space space space y equals 3 x plus 5 space space space space y minus 5 equals 3 x space space space space space space space 3 x equals y minus 5 space space space space space space space space space x equals fraction numerator y minus 5 over denominator 3 end fraction h to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus 5 over denominator 3 end fraction$

Sehingga:

$open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses g to the power of negative 1 end exponent open parentheses x close parentheses close parentheses open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses plus-or-minus square root of x plus 1 end root close parentheses open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals open parentheses f to the power of negative 1 end exponent close parentheses open parentheses fraction numerator plus-or-minus square root of x plus 1 end root minus 5 over denominator 3 end fraction close parentheses open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator negative 1 minus open parentheses plus-or-minus square root of x plus 1 end root close parentheses over denominator plus-or-minus square root of x plus 1 end root minus 2 end fraction$

Kemudian:

$space space space space space space space space open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator negative 1 minus open parentheses plus-or-minus square root of x plus 1 end root close parentheses over denominator plus-or-minus square root of x plus 1 end root minus 2 end fraction space space space space space space space space open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses a close parentheses equals fraction numerator negative 1 minus open parentheses plus-or-minus square root of a plus 1 end root close parentheses over denominator plus-or-minus square root of a plus 1 end root minus 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 2 equals fraction numerator negative 1 minus open parentheses plus-or-minus square root of a plus 1 end root close parentheses over denominator plus-or-minus square root of a plus 1 end root minus 2 end fraction space space space space space space space space space space space space space space 2 open parentheses plus-or-minus square root of a plus 1 end root minus 2 close parentheses equals negative 1 minus open parentheses plus-or-minus square root of a plus 1 end root close parentheses space space space space space space space space space space space space space space space space plus-or-minus 2 square root of a plus 1 end root minus 4 equals negative 1 minus open parentheses plus-or-minus square root of a plus 1 end root close parentheses space plus-or-minus 2 square root of a plus 1 end root plus open parentheses plus-or-minus square root of a plus 1 end root close parentheses equals 1 plus 4 space space space space space space space space space space space space space space space space space space space space space plus-or-minus 3 square root of a plus 1 end root equals 5 space space space space space space space space space space space space space space space space space space space space space space space space space space 9 open parentheses a plus 1 close parentheses equals 25 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a plus 1 equals 25 over 9 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a equals 25 over 9 minus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a equals fraction numerator 25 minus 9 over denominator 9 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space a equals 16 over 9$

Jadi, nilai adalah 16 over 9 .