Pertanyaan

Diketahui fungsi f ( x ) = 2 x + 4 dan g ( x ) = x − 4 2 x + 5 , x  = 4 . Jika h ( x ) = ( g ∘ f − 1 ) ( x ) dengan f − 1 ( x ) adalah invers fungsi f ( x ) dan h − 1 ( x ) adalah invers fungsi h ( x ) , fungsi h − 1 ( x ) = ...

Diketahui fungsi $f (x) = 2 x + 4$ dan $g (x) = \frac{2 x + 5}{x - 4}, x \neq = 4$ . Jika $h (x) = (g \circ f^{- 1}) (x)$ dengan $f^{- 1} (x)$ adalah invers fungsi $f (x)$ dan $h^{- 1} (x)$ adalah invers fungsi $h (x)$ , fungsi $h^{- 1} (x) =$ ...

$fraction numerator 12 x plus 2 over denominator x plus 2 end fraction comma space x not equal to negative 2$
$fraction numerator 12 x minus 2 over denominator x minus 2 end fraction comma space x not equal to 2$
$fraction numerator 12 x plus 2 over denominator x minus 2 end fraction comma space x not equal to 2$
$fraction numerator 6 x plus 2 over denominator x minus 2 end fraction comma space x not equal to 2$
$fraction numerator 6 x plus 2 over denominator x plus 2 end fraction comma space x not equal to negative 2$

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah C.

Pembahasan

Diketahui : , , , Cari terlebih dahulu. Misal sehingga Cari nilai Selanjutnya cari . Misal sehingga Oleh karena itu, jawaban yang benar adalah C.

Diketahui : f left parenthesis x right parenthesis equals 2 x plus 4 , $g left parenthesis x right parenthesis equals fraction numerator 2 x plus 5 over denominator x minus 4 end fraction comma space x not equal to 4$ , h left parenthesis x right parenthesis equals left parenthesis g ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis ,

Cari f to the power of negative 1 end exponent left parenthesis x right parenthesis terlebih dahulu. Misal y equals f left parenthesis x right parenthesis sehingga x equals f to the power of negative 1 end exponent left parenthesis y right parenthesis

$table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell 2 x plus 4 end cell row y equals cell 2 x plus 4 end cell row cell y minus 4 end cell equals cell 2 x end cell row cell fraction numerator y minus 4 over denominator 2 end fraction end cell equals x row x equals cell fraction numerator y minus 4 over denominator 2 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell fraction numerator y minus 4 over denominator 2 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator x minus 4 over denominator 2 end fraction end cell end table$

Cari nilai h left parenthesis x right parenthesis

$table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis end cell equals cell left parenthesis g ring operator f to the power of negative 1 end exponent right parenthesis left parenthesis x right parenthesis end cell row blank equals cell g left parenthesis f to the power of negative 1 end exponent left parenthesis x right parenthesis right parenthesis end cell row blank equals cell g left parenthesis fraction numerator x minus 4 over denominator 2 end fraction right parenthesis end cell row blank equals cell fraction numerator 2 left parenthesis fraction numerator x minus 4 over denominator 2 end fraction right parenthesis plus 5 over denominator left parenthesis begin display style fraction numerator x minus 4 over denominator 2 end fraction end style right parenthesis minus 4 end fraction end cell row blank equals cell fraction numerator x minus 4 plus 5 over denominator begin display style fraction numerator left parenthesis x minus 4 right parenthesis minus 8 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator 2 left parenthesis x plus 1 right parenthesis over denominator x minus 12 end fraction end cell row blank equals cell fraction numerator 2 x plus 2 over denominator x minus 12 end fraction end cell end table$

Selanjutnya cari h to the power of negative 1 end exponent left parenthesis x right parenthesis . Misal h left parenthesis x right parenthesis equals y sehingga x equals h to the power of negative 1 end exponent left parenthesis y right parenthesis

$table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x plus 2 over denominator x minus 12 end fraction end cell row y equals cell fraction numerator 2 x plus 2 over denominator x minus 12 end fraction end cell row cell y left parenthesis x minus 12 right parenthesis end cell equals cell 2 x plus 2 end cell row cell y x minus 12 y end cell equals cell 2 x plus 2 end cell row cell y x minus 2 x end cell equals cell 2 plus 12 y end cell row cell left parenthesis y minus 2 right parenthesis x end cell equals cell 12 y plus 2 end cell row x equals cell fraction numerator 12 y plus 2 over denominator y minus 2 end fraction end cell row cell h to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell fraction numerator 12 y plus 2 over denominator y minus 2 end fraction end cell row cell h to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 12 x plus 2 over denominator x minus 2 end fraction end cell end table$

Oleh karena itu, jawaban yang benar adalah C.