Pertanyaan

Diketahui fungsi f : R → R dan g : R → R . Jika fungsi f ( x ) = 2 x − 1 x + 2 , x > 2 1 dan g ( x ) = 2 x 2 + 1 x 2 , rumus fungsi ( g ∘ f ) − 1 ( x ) adalah...

Diketahui fungsi $f : R \to R$ dan $g : R \to R$ . Jika fungsi $f (x) = \frac{x + 2}{2 x - 1}, x > \frac{1}{2}$ dan $g (x) = \frac{x ^{2}}{2 x ^{2} + 1}$ , rumus fungsi $(g \circ f)^{- 1} (x)$ adalah...

$begin mathsize 14px style fraction numerator 3 x plus 2 over denominator 4 x minus 1 end fraction comma x not equal to 1 fourth end style$
$begin mathsize 14px style fraction numerator 3 x minus 2 over denominator 4 x minus 1 end fraction comma x not equal to 1 fourth end style$
$begin mathsize 14px style fraction numerator negative 3 x plus 2 over denominator 4 x minus 1 end fraction comma x not equal to 1 fourth end style$
$begin mathsize 14px style fraction numerator negative 3 x plus 2 over denominator 4 x minus 5 end fraction comma x not equal to 5 over 4 end style$
$begin mathsize 14px style fraction numerator negative 2 x plus 3 over denominator 4 x minus 1 end fraction comma x not equal to 1 fourth end style$

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah C.

Pembahasan

Dengan menggunakan konsep komposisi dan invers fungsi diperoleh ( g ∘ f ) ( x ) = = = = = = = = = g ( f ( x ) ) g ( 2 x − 1 x + 2 ) 2 ( 2 x − 1 x + 2 ) 2 + 1 ( 2 x − 1 x + 2 ) 2 2 ( 2 x − 1 x + 2 ) + 1 2 x − 1 x + 2 2 x − 1 2 x + 4 + 1 2 x − 1 x + 2 2 x − 1 2 x + 4 + 2 x − 1 2 x − 1 x + 2 2 x − 1 4 x + 3 2 x − 1 x + 2 2 x − 1 x + 2 ⋅ 4 x + 3 2 x − 1 4 x + 3 x + 2 Selanjutnya misalkan Dengan demikian . Oleh karena itu, jawaban yang benar adalah C .

Dengan menggunakan konsep komposisi dan invers fungsi diperoleh

$(g \circ f) (x) = = = = = = = = = g (f (x)) g (\frac{x + 2}{2 x - 1}) \frac{( \frac{x + 2}{2 x - 1} ) ^{2}}{2 ( \frac{x + 2}{2 x - 1} ) ^{2} + 1} \frac{\frac{x + 2}{2 x - 1}}{2 ( \frac{x + 2}{2 x - 1} ) + 1} \frac{\frac{x + 2}{2 x - 1}}{\frac{2 x + 4}{2 x - 1} + 1} \frac{\frac{x + 2}{2 x - 1}}{\frac{2 x + 4 + 2 x - 1}{2 x - 1}} \frac{\frac{x + 2}{2 x - 1}}{\frac{4 x + 3}{2 x - 1}} \frac{x + 2}{2 x - 1} \cdot \frac{2 x - 1}{4 x + 3} \frac{x + 2}{4 x + 3}$

Selanjutnya misalkan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals y row cell fraction numerator x plus 2 over denominator 4 x plus 3 end fraction end cell equals y row cell x plus 2 end cell equals cell y open parentheses 4 x plus 3 close parentheses end cell row cell x plus 2 end cell equals cell 4 x y plus 3 y end cell row cell x minus 4 x y end cell equals cell 3 y minus 2 end cell row cell x open parentheses 1 minus 4 y close parentheses end cell equals cell 3 y minus 2 end cell row x equals cell fraction numerator 3 y minus 2 over denominator 1 minus 4 y end fraction comma y not equal to 1 fourth end cell row x equals cell fraction numerator negative 3 y plus 2 over denominator 4 y minus 1 end fraction comma y not equal to 1 fourth end cell end table end style$

Dengan demikian $begin mathsize 14px style open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 3 x plus 2 over denominator 4 x minus 1 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank not equal to blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 fourth end cell end table end style$ .

Oleh karena itu, jawaban yang benar adalah C.