Diketahui f(x)=1+x1,x=0 dan (f∘g)(x)=x−12x−1,x=1. Nilai (g(2))⋅(g(3))⋅(g(4))⋅...⋅(g(99))⋅(g(100))=....

Pertanyaan

Diketahui $\style{font-size:14px}{f(x)=1+\frac1x,x\neq0}$ dan $\style{font-size:14px}{(f\circ g)(x)=\frac{2x-1}{x-1},x\neq1}$ . Nilai $\style{font-size:14px}{\left(g(2)\right)\cdot\left(g\left(3\right)\right)\cdot\left(g\left(4\right)\right)\cdot...\cdot\left(g\left(99\right)\right)\cdot\left(g\left(100\right)\right)=}$ ....

100
10
0,1
0,01
0,001

F. Ayudhita

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D

Pembahasan

$begin mathsize 14px style left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals fraction numerator 2 x minus 1 over denominator x minus 1 end fraction left parenthesis f left parenthesis g left parenthesis x right parenthesis right parenthesis equals fraction numerator 2 x minus 1 over denominator x minus 1 end fraction end style$

Masukan ke , sehingga:

$size 14px 1 size 14px plus fraction numerator size 14px 1 over denominator size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis end fraction size 14px equals fraction numerator size 14px 2 size 14px x size 14px minus size 14px 1 over denominator size 14px x size 14px minus size 14px 1 end fraction size 14px 1 size 14px plus fraction numerator size 14px 1 over denominator size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis end fraction size 14px minus size 14px 1 size 14px equals fraction numerator size 14px 2 size 14px x size 14px minus size 14px 1 over denominator size 14px x size 14px minus size 14px 1 end fraction size 14px minus size 14px 1 fraction numerator size 14px 1 over denominator size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis end fraction size 14px equals fraction numerator size 14px 2 size 14px x size 14px minus size 14px 1 over denominator size 14px x size 14px minus size 14px 1 end fraction size 14px minus fraction numerator size 14px x size 14px minus size 14px 1 over denominator size 14px x size 14px minus size 14px 1 end fraction fraction numerator size 14px 1 over denominator size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis end fraction size 14px equals fraction numerator size 14px 2 size 14px x size 14px minus size 14px 1 size 14px minus size 14px left parenthesis size 14px x size 14px minus size 14px 1 size 14px right parenthesis over denominator size 14px x size 14px minus size 14px 1 end fraction fraction numerator size 14px 1 over denominator size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis end fraction size 14px equals fraction numerator size 14px 2 size 14px x size 14px minus size 14px 1 size 14px minus size 14px x size 14px plus size 14px 1 over denominator size 14px x size 14px minus size 14px 1 end fraction fraction numerator size 14px 1 over denominator size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis end fraction size 14px equals fraction numerator size 14px x over denominator size 14px x size 14px minus size 14px 1 end fraction size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis size 14px equals fraction numerator size 14px x size 14px minus size 14px 1 over denominator size 14px x end fraction$

Maka dapat kita peroleh:

$begin mathsize 14px style left parenthesis fraction numerator 2 minus 1 over denominator 2 end fraction right parenthesis end style size 14px times begin mathsize 14px style open parentheses fraction numerator 3 minus 1 over denominator 3 end fraction close parentheses end style size 14px times begin mathsize 14px style left parenthesis fraction numerator 4 minus 1 over denominator 4 end fraction right parenthesis end style size 14px times size 14px. size 14px. size 14px. size 14px times begin mathsize 14px style left parenthesis fraction numerator 99 minus 1 over denominator 99 end fraction right parenthesis end style size 14px times begin mathsize 14px style left parenthesis fraction numerator 100 minus 1 over denominator 100 end fraction right parenthesis end style size 14px equals begin mathsize 14px style left parenthesis 1 half right parenthesis end style size 14px times begin mathsize 14px style left parenthesis 2 over 3 right parenthesis end style size 14px times begin mathsize 14px style open parentheses 3 over 4 close parentheses end style size 14px times size 14px. size 14px. size 14px. size 14px times begin mathsize 14px style left parenthesis 98 over 99 right parenthesis end style size 14px times begin mathsize 14px style open parentheses 99 over 100 close parentheses end style size 14px equals begin mathsize 14px style left parenthesis fraction numerator 1 times left parenthesis 2 times 3 times... times 98 times 99 right parenthesis over denominator left parenthesis 2 times 3 times 4 times... times 99 right parenthesis times 100 end fraction right parenthesis end style size 14px equals size 14px 1 over size 14px 100 size 14px equals size 14px 0 size 14px comma size 14px 01$

Jadi, jawaban yang tepat adalah D