Question:

<div class="solution"><p><strong>Key Idea:</strong> Complete the square. Domain $x\ge\frac{7}{2}$ tells which root to keep.</p><p><strong>Step 1:</strong> $y = (x-\frac{7}{2})^2 - \frac{9}{4}$</p><p><strong>Step 2:</strong> $x-\frac{7}{2} = +\sqrt{y+\frac{9}{4}}$ (positive, since $x\ge\frac{7}{2}$)</p><p><strong>Step 3:</strong> <span class="math-block">$$f^{-1}(x) = \frac{7+\sqrt{9+4x}}{2}$$</p><div class="trap-box"><strong>Trap:</strong> If you keep both \pm signs, the inverse is not a function. The domain restriction removes one branch.<div class="key-concept"><strong>Key Concept:</strong> Inverse of restricted parabola -- domain decides which square-root branch survives