Pertanyaan

Diketahui f ( x ) = x + 2 5 x − 3 , x  = − 2 dan g ( x ) = 6 x − 2 , tentukan: a. f − 1 ( x ) dan g − 1 ( x ) b. ( f ∘ g ) − 1 ( x ) dan ( g ∘ f ) − 1 ( x )

Diketahui $f (x) = \frac{5 x - 3}{x + 2}, x \neq = - 2$ dan $g (x) = 6 x - 2$ , tentukan:

a. $f^{- 1} (x)$ dan $g^{- 1} (x)$

b. $(f \circ g)^{- 1} (x)$ dan $(g \circ f)^{- 1} (x)$

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R. Septa

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

a. , inversnya adalah dan, , maka inversnya adalah b. Invers dari fungsi komposisinya adalah sebagai berikut Dan,

a. $f open parentheses x close parentheses equals fraction numerator 5 x minus 3 over denominator x plus 2 end fraction comma space x not equal to negative 2$ , inversnya adalah

$table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator 5 x minus 3 over denominator x plus 2 end fraction end cell row cell x y plus 2 y end cell equals cell 5 x minus 3 end cell row cell x y minus 5 x end cell equals cell negative 2 y minus 3 end cell row cell x open parentheses y minus 5 close parentheses end cell equals cell negative 2 y minus 3 end cell row x equals cell fraction numerator negative 2 y minus 3 over denominator y minus 5 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction comma space x not equal to 5 end cell end table$

dan, g open parentheses x close parentheses equals 6 x minus 2 , maka inversnya adalah

$y equals 6 x minus 2 6 x equals y plus 2 x equals fraction numerator y plus 2 over denominator 6 end fraction g to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x plus 2 over denominator 6 end fraction$

b. Invers dari fungsi komposisinya adalah sebagai berikut

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction close parentheses end cell row blank equals cell fraction numerator fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction plus 2 over denominator 6 end fraction end cell row blank equals cell fraction numerator fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction plus 2 over denominator 6 end fraction cross times fraction numerator x minus 5 over denominator x minus 5 end fraction end cell row blank equals cell fraction numerator negative 2 x minus 3 plus 2 x minus 10 over denominator 6 x minus 30 end fraction end cell row blank equals cell negative fraction numerator 13 over denominator 6 x minus 30 end fraction comma space x not equal to 5 end cell end table$

Dan,

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses fraction numerator x plus 2 over denominator 6 end fraction close parentheses end cell row blank equals cell fraction numerator negative 2 open parentheses fraction numerator x plus 2 over denominator 6 end fraction close parentheses minus 3 over denominator fraction numerator x plus 2 over denominator 6 end fraction minus 5 end fraction end cell row blank equals cell fraction numerator negative 2 open parentheses fraction numerator x plus 2 over denominator 6 end fraction close parentheses minus 3 over denominator fraction numerator x plus 2 over denominator 6 end fraction minus 5 end fraction cross times 6 over 6 end cell row blank equals cell fraction numerator negative 2 x minus 4 minus 18 over denominator x plus 2 minus 30 end fraction end cell row blank equals cell fraction numerator negative 2 x minus 22 over denominator x minus 28 end fraction comma space x not equal to 28 end cell end table$

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