Diketahui f(x)=x+25x−3, x=−2 dan g(x)=6x−2, tentukan: a. f−1(x) dan g−1(x) b. (f∘g)−1(x) dan (g∘f)−1(x)

Pertanyaan

Diketahui $f open parentheses x close parentheses equals fraction numerator 5 x minus 3 over denominator x plus 2 end fraction comma space x not equal to negative 2$ dan $g\left(x\right)=6x-2$ , tentukan:

a. $f^{-1}\left(x\right)$ dan $g^{-1}\left(x\right)$

b. $\left(f\circ g\right)^{-1}\left(x\right)$ dan $\left(g\circ f\right)^{-1}\left(x\right)$

R. Septa

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

a. $f open parentheses x close parentheses equals fraction numerator 5 x minus 3 over denominator x plus 2 end fraction comma space x not equal to negative 2$ , inversnya adalah

$table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator 5 x minus 3 over denominator x plus 2 end fraction end cell row cell x y plus 2 y end cell equals cell 5 x minus 3 end cell row cell x y minus 5 x end cell equals cell negative 2 y minus 3 end cell row cell x open parentheses y minus 5 close parentheses end cell equals cell negative 2 y minus 3 end cell row x equals cell fraction numerator negative 2 y minus 3 over denominator y minus 5 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction comma space x not equal to 5 end cell end table$

dan, g open parentheses x close parentheses equals 6 x minus 2 , maka inversnya adalah

$y equals 6 x minus 2 6 x equals y plus 2 x equals fraction numerator y plus 2 over denominator 6 end fraction g to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x plus 2 over denominator 6 end fraction$

b. Invers dari fungsi komposisinya adalah sebagai berikut

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction close parentheses end cell row blank equals cell fraction numerator fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction plus 2 over denominator 6 end fraction end cell row blank equals cell fraction numerator fraction numerator negative 2 x minus 3 over denominator x minus 5 end fraction plus 2 over denominator 6 end fraction cross times fraction numerator x minus 5 over denominator x minus 5 end fraction end cell row blank equals cell fraction numerator negative 2 x minus 3 plus 2 x minus 10 over denominator 6 x minus 30 end fraction end cell row blank equals cell negative fraction numerator 13 over denominator 6 x minus 30 end fraction comma space x not equal to 5 end cell end table$

Dan,

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses fraction numerator x plus 2 over denominator 6 end fraction close parentheses end cell row blank equals cell fraction numerator negative 2 open parentheses fraction numerator x plus 2 over denominator 6 end fraction close parentheses minus 3 over denominator fraction numerator x plus 2 over denominator 6 end fraction minus 5 end fraction end cell row blank equals cell fraction numerator negative 2 open parentheses fraction numerator x plus 2 over denominator 6 end fraction close parentheses minus 3 over denominator fraction numerator x plus 2 over denominator 6 end fraction minus 5 end fraction cross times 6 over 6 end cell row blank equals cell fraction numerator negative 2 x minus 4 minus 18 over denominator x plus 2 minus 30 end fraction end cell row blank equals cell fraction numerator negative 2 x minus 22 over denominator x minus 28 end fraction comma space x not equal to 28 end cell end table$

130

4.7 (11 rating)