<p>If \(f:\mathbb{R}\setminus\{1,-1\}\to A\), \(f(x)=\dfrac{x^2}{1-x^2}\) is surjective, find \(A\).</p>
Step-by-Step Solution
<div class="solution"><p>Let t=x^2\ge0, t≠1. For 0\let<1: y=t/(1-t)\in[0,\infty). For t>1: y\in(-\infty,-1). So A=(-\infty,-1)\cup[0,\infty).</p><p><strong>Answer: $A=(-\infty,-1)\cup[0,\infty)$</strong></p><div class="trap-box"><strong>Trap:</strong> x^2\ge0, so values from (-1,0) never occur.<div class="key-concept"><strong>Key Concept:</strong> Rational function in x^2 \to substitute t=x^2\ge0
Correct Answer: A = (-∞,-1)∪[0,∞)