<p>If <span class="math-inline">\(f:\mathbb{R}\setminus\{1,-1\}\to A\)</span>, <span class="math-inline">\(f(x)=\dfrac{x^2}{1-x^2}\)</span> is surjective, find <span class="math-inline">\(A\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Let t=x²≥0, t≠1. For 0≤t<1: y=t/(1-t)∈[0,∞). For t>1: y∈(-∞,-1). So A=(-∞,-1)∪[0,∞).</p><p><strong>Answer: <span class="math-inline">$A=(-\infty,-1)\cup[0,\infty)$</span></strong></p><div class="trap-box"><strong>Trap:</strong> x²≥0, so values from (-1,0) never occur.</div><div class="key-concept"><strong>Key Concept:</strong> Rational function in x² → substitute t=x²≥0</div></div>
Correct Answer: A = (-∞,-1)∪[0,∞)