Question:

<div class="solution"><p>y=(x+1)^2+2 \to x+1=\pm\sqrt(y-2). Since x\le-2, x+1\le-1<0 \to negative root: x=-1-\sqrt(y-2).</p><p><span class="math-block">$$f^{-1}(x) = -\sqrt{x-2}-1$$</p><div class="trap-box"><strong>Trap:</strong> Positive root violates the restricted domain.<div class="key-concept"><strong>Key Concept:</strong> Inverse of restricted parabola -- domain determines root sign