<p>Consider <span class="math-inline">\(f(x) = \frac{|x+1|}{x+1} + \frac{x-1}{|x-1|}\)</span>. Which are correct?</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Each fraction is a sign function. Evaluate on three intervals (excluding x=±1):</p><p>x<-1: f=-1-1=-2; -1<x<1: f=1-1=0; x>1: f=1+1=2</p><p>Range={-2,0,2}: bounded ✓, 3 integers ✓, many-one ✓</p><p>f(-x)=-f(x): odd ✓</p><p><strong>Answer: All four (A),(B),(C),(D)</strong></p><div class="trap-box"><strong>Trap:</strong> Domain excludes x=±1. Do not assign values there.</div><div class="key-concept"><strong>Key Concept:</strong> |u|/u = sgn(u) — sign analysis gives discrete range</div></div>
Correct Answer: A,B,C,D