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Turkey NMO 2000 1st Round - P28 (Algebra)

Source:

July 25, 2012
function

Problem Statement

f1(x)=x2+xf2(x)=2x2−xf3(x)=x2+xg1(x)=x−2g2(x)=2x  g3(x)=x+2\begin{array}{ rlrlrl} f_1(x)=&x^2+x & f_2(x)=&2x^2-x & f_3(x)=&x^2 +x \\ g_1(x)=&x-2 & g_2(x)=&2x \ \ & g_3(x)=&x+2 \\ \end{array} If h(x)=xh(x)=x can be get from fif_i and gig_i by using only addition, substraction, multiplication defined on those functions where i∈{1,2,3}i\in\{1,2,3\}, then Fi=1F_i=1. Otherwise, Fi=0F_i=0. What is (F1,F2,F3)(F_1,F_2,F_3) ?
<spanclass=′latex−bold′>(A)</span> (0,0,0)<spanclass=′latex−bold′>(B)</span> (0,0,1)<spanclass=′latex−bold′>(C)</span> (0,1,0)<spanclass=′latex−bold′>(D)</span> (0,1,1)<spanclass=′latex−bold′>(E)</span> None <span class='latex-bold'>(A)</span>\ (0,0,0) \qquad<span class='latex-bold'>(B)</span>\ (0,0,1) \qquad<span class='latex-bold'>(C)</span>\ (0,1,0) \qquad<span class='latex-bold'>(D)</span>\ (0,1,1) \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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