Question:

<div class="solution"><p><strong>Key Idea:</strong> Test onto via discriminant, test one-one via derivative sign changes.</p><p><strong>Step 1:</strong> Denominator is never zero: $\Delta = 81 - 352 < 0$. Domain is all $\mathbb{R}$.</p><p><strong>Step 2 (Onto):</strong> Set $y = f(x)$, rearrange to quadratic in $x$. Discriminant condition gives $1 + 226y - 271y^2 \ge 0$ -- not true for all real $y$. So <strong>not onto</strong>.</p><p><strong>Step 3 (One-one):</strong> $f'(x)$ has real roots, so $f'$ changes sign. So <strong>not one-one</strong>.</p><p><strong>Answer: Many-one into</strong></p><div class="trap-box"><strong>Trap:</strong> Having domain $\mathbb{R}$ does not imply range $\mathbb{R}$. Equal-degree rational functions often miss an interval.<div class="key-concept"><strong>Key Concept:</strong> Range of rational function via discriminant; injectivity via derivative