Sets, Relations & Functions
General
Grade 11
Question:
<p>Let <span class="math-inline">\(f:(-\infty,-2]\to[3,\infty)\)</span>, <span class="math-inline">\(f(x)=x^2+2x+3\)</span>. Find <span class="math-inline">\(f^{-1}(x)\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>y=(x+1)²+2 → x+1=±√(y-2). Since x≤-2, x+1≤-1<0 → negative root: x=-1-√(y-2).</p><p><span class="math-block">$$f^{-1}(x) = -\sqrt{x-2}-1$$</span></p><div class="trap-box"><strong>Trap:</strong> Positive root violates the restricted domain.</div><div class="key-concept"><strong>Key Concept:</strong> Inverse of restricted parabola — domain determines root sign</div></div>
Correct Answer: -√(x-2) - 1