Pertanyaan

Diketahui f(x) = 2x + 3 dengan dan g(x) = 3x - 2 dengan . Maka, ...

Diketahui f(x) = 2x + 3 dengan dan g(x) = 3x - 2 dengan . Maka,...

$begin mathsize 14px style fraction numerator x plus 1 over denominator 6 end fraction end style$
$begin mathsize 14px style fraction numerator x plus 2 over denominator 3 end fraction end style$
$begin mathsize 14px style fraction numerator x minus 1 over denominator 6 end fraction end style$
$begin mathsize 14px style fraction numerator x minus 3 over denominator 2 end fraction end style$
$begin mathsize 14px style fraction numerator x minus 7 over denominator 6 end fraction end style$

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Klaim

A. Rizky

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

Dapat diperiksa bahwa (f ∘ g) (x) terdefinisi dan inversnya ada, serta masing-masing f dan g memiliki invers. Pertama, dengan menerapkan sifat invers pada fungsi komposisi, maka Selanjutnya, kita cari masing-masing dan . Dengan f(x) = y, maka (y) = x sehingga Dengan g(x) = y, maka (y) = x sehingga Jadi,

Dapat diperiksa bahwa (f ∘ g) (x) terdefinisi dan inversnya ada, serta masing-masing f dan g memiliki invers.

Pertama, dengan menerapkan sifat invers pada fungsi komposisi, maka

Selanjutnya, kita cari masing-masing dan .

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses x close parentheses end cell equals cell 2 x plus 3 end cell row y equals cell 2 x plus 3 end cell row cell y minus 3 end cell equals cell 2 x end cell row cell fraction numerator y minus 3 over denominator 2 end fraction end cell equals x end table end style$

Dengan f(x) = y, maka (y) = x sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals x row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator y minus 3 over denominator 2 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator x minus 3 over denominator 2 end fraction end cell row blank blank blank row cell g open parentheses x close parentheses end cell equals cell 3 x minus 2 end cell row y equals cell 3 x minus 2 end cell row cell y plus 2 end cell equals cell 3 x end cell row cell fraction numerator y plus 2 over denominator 3 end fraction end cell equals x end table end style$

Dengan g(x) = y, maka (y) = x sehingga

$begin mathsize 14px style g to the power of negative 1 end exponent open parentheses y close parentheses equals x g to the power of negative 1 end exponent open parentheses y close parentheses equals fraction numerator y plus 2 over denominator 3 end fraction g to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x plus 2 over denominator 3 end fraction end style$

Jadi,

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