Pertanyaan

Diketahui f ( x ) = 3 x + 1 dan g ( x ) = x + 3 2 x − 1 , x  = − 3 . Jika invers ( g ∘ f ) ( x ) adalah ( g ∘ f ) − 1 ( x ) , maka ( g ∘ f ) − 1 ( − 2 ) adalah ...

Diketahui $f (x) = 3 x + 1$ dan $g (x) = \frac{2 x - 1}{x + 3}, x \neq = - 3$ . Jika invers $(g \circ f) (x)$ adalah $(g \circ f)^{- 1} (x)$ , maka $(g \circ f)^{- 1} (- 2)$ adalah ...

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

dan , maka Jadi , misalkan , maka Jadi, invers dari adalah , sehingga Jadi, jawaban yang tepat adalah C.

begin mathsize 14px style f open parentheses x close parentheses equals 3 x plus 1 end style dan $begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 3 end fraction comma space x not equal to negative 3 end style$ , maka

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell g open parentheses 3 x plus 1 close parentheses end cell row blank equals cell fraction numerator 2 open parentheses 3 x plus 1 close parentheses minus 1 over denominator open parentheses 3 x plus 1 close parentheses plus 3 end fraction end cell row blank equals cell fraction numerator 6 x plus 2 minus 1 over denominator 3 x plus 4 end fraction end cell row blank equals cell fraction numerator 6 x plus 1 over denominator 3 x plus 4 end fraction end cell end table end style$

Jadi $begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 6 x plus 1 over denominator 3 x plus 4 end fraction end style$ , misalkan , maka

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell fraction numerator 6 x plus 1 over denominator 3 x plus 4 end fraction end cell row y equals cell fraction numerator 6 x plus 1 over denominator 3 x plus 4 end fraction end cell row cell y open parentheses 3 x plus 4 close parentheses end cell equals cell 6 x plus 1 end cell row cell 3 x y plus 4 y end cell equals cell 6 x plus 1 end cell row cell 3 x y minus 6 x end cell equals cell 1 minus 4 y end cell row cell x open parentheses 3 y minus 6 close parentheses end cell equals cell 1 minus 4 y end cell row x equals cell fraction numerator 1 minus 4 y over denominator 3 y minus 6 end fraction end cell row x equals cell fraction numerator negative 4 y plus 1 over denominator 3 y minus 6 end fraction end cell row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 4 x plus 1 over denominator 3 x minus 6 end fraction end cell end table end style$

Jadi, invers dari begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses end style adalah $begin mathsize 14px style open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative 4 x plus 1 over denominator 3 x minus 6 end fraction end style$ , sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses negative 2 close parentheses end cell equals cell fraction numerator negative 4 open parentheses negative 2 close parentheses plus 1 over denominator 3 open parentheses negative 2 close parentheses minus 6 end fraction end cell row blank equals cell fraction numerator 8 plus 1 over denominator negative 6 minus 6 end fraction end cell row blank equals cell fraction numerator 9 over denominator negative 12 end fraction end cell row blank equals cell negative 3 over 4 end cell end table end style$

Jadi, jawaban yang tepat adalah C.