Pertanyaan

Diketahui f(x) = 2x - 5 dan dengan . Maka, adalah ....

Diketahui f(x) = 2x - 5 dan $begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator 2 over denominator x plus 1 end fraction end style$ dengan . Maka, adalah ....

$begin mathsize 14px style negative fraction numerator 5 plus x over denominator x plus 1 end fraction end style$
$begin mathsize 14px style fraction numerator 1 plus x over denominator x plus 5 end fraction end style$
$begin mathsize 14px style fraction numerator 5 plus x over denominator x plus 1 end fraction end style$
$begin mathsize 14px style negative fraction numerator 1 plus x over denominator x plus 5 end fraction end style$
$begin mathsize 14px style negative fraction numerator 1 over denominator x plus 5 end fraction end style$

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Dapat diperiksa bahwa (f∘g) (x) terdefinisi dan inversnya ada serta masing-masing f dan g punya invers. Pertama, kita cari masing-masing fungsi invers dari f dan g. Dengan f(x) = y, maka Dengan g(x) = y, maka Sehingga Jadi, jawaban yang tepat adalah D.

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

Pertama, kita cari masing-masing fungsi invers dari f dan g.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell 2 x minus 5 end cell row y equals cell 2 x minus 5 end cell row cell y plus 5 end cell equals cell 2 x end cell row cell fraction numerator y plus 5 over denominator 2 end fraction end cell equals x end table end style$

Dengan f(x) = y, maka

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals x row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator y plus 5 over denominator 2 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator x plus 5 over denominator 2 end fraction end cell row blank blank blank row cell g open parentheses x close parentheses end cell equals cell fraction numerator 2 over denominator x plus 1 end fraction end cell row y equals cell fraction numerator 2 over denominator x plus 1 end fraction end cell row cell y open parentheses x plus 1 close parentheses end cell equals 2 row cell x y plus y end cell equals 2 row cell x y end cell equals cell 2 minus y end cell row x equals cell fraction numerator 2 minus y over denominator y end fraction end cell row x equals cell 2 over y minus 1 end cell end table end style$

Dengan g(x) = y, maka

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 2 over y minus 1 g to the power of negative 1 end exponent open parentheses x close parentheses equals 2 over x minus 1 end style

Sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator x plus 5 over denominator 2 end fraction close parentheses end cell row blank equals cell fraction numerator 2 over denominator open parentheses fraction numerator x plus 5 over denominator 2 end fraction close parentheses end fraction minus 1 end cell row blank equals cell fraction numerator 2 times 2 over denominator x plus 5 end fraction minus 1 end cell row blank equals cell fraction numerator 4 over denominator x plus 5 end fraction minus 1 end cell row blank equals cell fraction numerator 4 over denominator x plus 5 end fraction minus fraction numerator x plus 5 over denominator x plus 5 end fraction end cell row blank equals cell fraction numerator negative 1 minus x over denominator x plus 5 end fraction end cell row blank equals cell negative fraction numerator 1 plus x over denominator x plus 5 end fraction end cell end table end style$

Jadi, jawaban yang tepat adalah D.