Functions
General
Grade 12
Question:
Let f(x) = \frac{x}{1-x} and let \alpha be a real number. If x_0 = \alpha, x_1 = f(x_0), x_2 = f(x_1), \ldots and x_{2011} = -\frac{1}{2012} then the value of \alpha is
Step-by-Step Solution
<div class="solution"><p>(A) f(3/2)=9/4; f(9/4)=9/2-3=3/2; f(3/2)=9/4 ✓</p><p>(B) f(5/2)=2; f(2)=1; f(1)=1; so 1+1=2=f(5/2) ✓</p><p>(C) f(1)=1; f(f(1))=f(1)=1 ✓; (D) false.</p><p><strong>Answer: (A),(B),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> Output of one branch can fall into a different branch on next iteration. Re-check interval each time.</div><div class="key-concept"><strong>Key Concept:</strong> Piecewise iteration — branch-tracing, slow and exact</div></div>
Correct Answer: D